Abstract:
The invention provides a system and method for limited view ultrasound imaging of a 2D section or a 3D volume of a body part. Ultrasound sensors configured are spatially or temporally arrayed in a limited view circular arc or over at least part of a concave surface such as a hemisphere. A processor calculates from detected ultrasound radiation a beam forming (BF) functional and calculates from the free amplitudes a point spread function (PSF). A filter g(k) is calculated from the Fourier transform H BF (k) of the PSF that is used to generate an image of the 2D section or the 3D volume of the body part.

Description:
PRIORITY INFORMATION 
       [0001]    This National Stage application claims priority to U.S. Provisional Application No. 61/420,098 filed on Dec. 6, 2010. 
     
    
     FIELD OF THE INVENTION 
       [0002]    This invention relates to medical devices and more specifically to such devices for medical imaging by ultrasound. 
       BACKGROUND OF THE INVENTION 
       [0003]    The following prior art publications are considered to be relevant for an understanding of the prior art.
   1. Report on “Mammography and beyond: developing technologies for the early detection of breast cancer”,   2. U.S.A. Institute of Medicine and the Governing Board of the National Research Council, National Academy Press, 2101 Constitution Avenue, N.W., Box 285, Washington, D.C. 20055.   3. Chandra M. Sehgal et al, Journal of Mammary Gland Biology and Neoplasia, Volume 11, Number 2, April, 2006.   4. W. E. Svensso, “Breast Ultrasound Update”, ULTRASOUND, February 2006, Volume 14, Number 1, 20-33.   5. J. F. Greenleaf et al on quantitative cross-sectional imaging of ultrasound parameters, Ultrasonics Symposium Proc., IEEE Cat. #77CH1264-1SU, pp. 989-995, 1977.   6. Littrup PJ et al Proceedings of the 26th International Acoustical Imaging Symposium, Windsor, Canada; Sep. 9-12, 2001.   7. An early patent issued in 1978 by Robert E. Frazer “Coupling apparatus for ultrasonic medical diagnostic system”, U.S. Pat. No. 4,130,112.   8. US patent Publication No. 20060241423 to Tor C Anderson et al.   9. US Patent publication 20060173307.   10. US patent publication 20070055159.   11. Thomas R. Nelson et al Proceedings of SPIE—Volume 6510, March 2007.   12. U.S. Pat. No. 4,509,368.   13. U.S. Pat. No. 7,025,725.   14. U.S. Pat. No. 7,264,592.   15. Karmanos Cancer Institute, Lawrence Livermore National Laboratory report [UCRL-JRNL-207220] of April 2004.   16. Lawrence Livermore National Laboratory report [UCRL-JRNL-207220] of April 2004.   17. International Patent Publication WO03/103500.   18. US patent publication number 20060241423.   19. U.S. Pat. No. 5,660,185.   20. U.S. Pat. No. 5,664,573.   21. A. Fenster et al, Ultrasound Med Biol. August; 27(8):1025-34, 2001.   22. J. F. Greenleaf et al Ultrasonics Symposium Proc., IEEE Cat. #77CH1264-1SU, pp. 989-995, 1977.   23. US patent publication No. 20060241423.   24. US Patent publication 20060173307.   25. US patent publication 20070055159.   26. U.S. Pat. No. 4,509,368.   27. U.S. Pat. No. 7,025,725.   28. U.S. Pat. No. 7,264,592.   29. U.S. Pat. No. 5,660,185.   30. U.S. Pat. No. 5,664,573.   31. Simonetti, F. &amp; Huang, L. 2008, “From beamforming to diffraction tomography”, J. Appl. Phys. 103, 103 110.   32. US Patent Publication 2006/0009693.   33. Devaney, A. J. 1982, “A filtered backpropagation algorithm for diffraction tomography”, Ultrason. Imaging 4, 336-350.   34. Born, M. &amp; Wolf, E. 1999 Principles of optics. Cambridge, UK: Cambridge University Press.   35. Kak, A. C. &amp; Slaney, M. 1988 Principles of computerized tomographic imaging. New York, N.Y.: IEEE Press   36. Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962, p. 331 and 332.   37. http://en.wikipedia.org/wiki/Legendre_function.   38. http://en.wikipedia.org/wiki/Hypergeometric_function.   39. http://en.wikipedia.org/wiki/Gamma_function.   40. http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/07/07/01/   41. http://en.wikipedia.org/wiki/Bessel_function   42. http://en.wikipedia.org/wiki/JacobiAnger_expansion   43. http://en.wikipedia.org/wiki/Angle_addition_formula#Angle_sum_and_difference_id entities   44. http://farside.ph.utexas.edu/teaching/jk1/lectures/node102.html (Spherical Harmonics)   45. W. Jarosz, N. Carr &amp; H. W. Jensen, “Importance Sampling Spherical Harmonics”, Journal compilation, 2008, The Eurographics Association and Blackwell Publishing Ltd.   46. R Mehremt, J T Londergant and M H Macfarlanet, “Analytic expressions for integrals of products of spherical Bessel functions”, J. Phys. A: Math. Gen. 24(1991) 1435-1453.   47. Edmonds A R 1957 Angular Momentum in Quantum Mechanics (Princeton: Princeton University Academic Press   48. M. R. Aguilar, C. Elvira, A. Gallardo, B. Vazquez, and J. S. Roman, “Smart polymers and Their Applications as Biomaterials”, Topics in Tissue Engineering, Vol. 3, 2007. Eds. N Ashammakhi, R Reis &amp; E Chiellini© 2007.   49. S. F. Khattak, S. R. Bhatia, S. C. Roberts, “Pluronic F127 as a Cell Encapsulation Material: Utilization of Membrane-Stabilizing Agents”, Tissue Engineering Volume 11, Number 5/6, 2005   
 
         [0053]    Breast cancer is one of the leading causes of death from cancer. Early detection is widely believed to reduce breast cancer mortality by allowing intervention at an earlier stage of cancer progression. Screening [X-ray] mammography has secured a place as the gold standard routine health maintenance procedure for women—this is a mature technology that provides high-quality images in the majority of patients. 
         [0054]    However, conventional mammography does not detect all breast cancers, including some that are palpable, and as many as three-quarters of all breast lesions biopsied because of a suspicious finding on a mammogram turn out to be benign. 
         [0055]    Mammograms are particularly difficult to interpret for women with dense breast tissue, who are at increased risk of breast cancer. The dense tissue interferes with the identification of abnormalities associated with tumors. Accordingly, other imaging technologies, particularly non-ionizing modalities such as magnetic resonance imaging and ultrasound, are being tested for application to breast cancer. These methods may provide additional diagnostic specificity over X-ray mammography alone. These include ultrasound examinations, and needle or surgical biopsies. Improvements in these techniques are desirable for more accurate and less invasive treatment. 
         [0056]    Ultrasound examinations are routinely done following a suspicious finding, for example in order to discriminate a cyst from a solid mass. Furthermore, for women with dense breast tissue, ultrasound is used even for screening. However, the varied, heterogeneous, complex structure of the breast makes good breast ultrasound imaging more difficult than in many other regions of the body. Conventional ultrasound has a limited field of view, is not reproducible and produces results that are a trade-off between penetration and image resolution. It is generally perceived that ultrasound is incapable of reliably detecting micro-calcifications, these being early indications of breast cancer. Conventional two dimensional [2D] ultrasound procedures dynamically distort the breast during the examination, thereby making it difficult to determine the exact location of tumors or other masses. The 2D nature of conventional ultrasound and the distortion of the breast during the examination, make it difficult to guide biopsy or ablation procedures. 
         [0057]    Several approaches to introducing 3D breast ultrasound procedures have been attempted in the prior art. These include conventional 3D scanning as well as Ultrasound Computed Tomography. 
         [0058]    The conventional 3D scanning approach is driven by the need for real-time data acquisition and display. Therefore, some of the complex physics associated with propagation of sound waves is traded off. One of the tradeoffs corresponds to the usage of straight-ray theory, a basic approximation of the true physics of acoustic wave propagation, which is only valid for purely homogenous media. A second important tradeoff is the assumption of a 2D geometry in which only the directly backscattered reflections are collected. In reality, the emitted pulses interact so strongly with the tissue that a “scattered field” of acoustic energy is produced and acoustic waves are distributed in all directions. This means that in conventional 3D ultrasound, only a small fraction of the scattered waves arrive at the detectors. Consequently, the strength of the detected signal is weak and it is necessary to “beam form” (focus energy in a specific direction) in order to amplify the return signal. 3D and also four-dimensional [4D] standard transducers are also used for breast imaging. The main motivation behind their usage is the need for real-time data acquisition and display. As with 2D conventional ultrasound, the distortion of the breast during the examination with such transducers disturbs the determination of the exact location of tumors or other masses and makes it difficult to guide biopsy or ablation procedures. In addition, as in conventional 2D ultrasound, these transducers detect back scattered sound waves only, thus lacking the advantages introduced by tomography methods discussed below. 
         [0059]    A tomography approach may “undo” these trade-offs, leading to a marked increase in the signal-to-noise ratio, while reducing artifacts and yielding higher quality images for greater clinical sensitivity. Furthermore, the signals that propagate through the breast, and which are never reflected back, contain additional information. These transmitted signals can be used to calculate acoustic parameters, not contained in the reflection data, such as sound speed and attenuation, possibly leading to greater clinical specificity. Prior art contributions to this conclusion include the pioneering paper by J. F. Greenleaf et al. [22]. 
         [0060]    One of a variety of obstacles to developing a device capable of facilitating scanning of the whole breast volume in a short time is the difficulty of maintaining good contact between the ultrasound transducer and the skin and the image quality that results from a controlled but inflexible scanning mechanism. 
         [0061]    A full-field breast ultrasound (FFBU) scanning apparatus and related methods were suggested in the prior art. For example, US patent Publication: 20060241423 of Anderson et al. discloses such a device. However, this device required the compression of the breast. 
         [0062]    US Patent Publication 20060173307 of Amara Arie et al describes a circular breast ultrasound scanner (CBUS). The CBUS system automatically images the complete breast. A mechanical device for moving a standard 2D ultrasound transducer in a circular or helical motion is used. The breast is placed in a hollow [vacuum] housing. 
         [0063]    US Patent Publication 20070055159 to Wang Shih-Ping et al, describes an apparatus and related methods for facilitating volumetric ultrasonic scanning of a breast. A cone-shaped radial scanning template is rotated, thereby moving an ultrasound transducer to scan the breast. A flexible membrane is mounted on a mechanical assembly to form a slot-like opening through which an ultrasound transducer directly contacts the skin surface. The breast in this procedure has to be compressed. 
         [0064]    The publication of Thomas R. Nelson et al [11] discloses a volume breast ultrasound scanner that images a pendant breast. Their performance assessment characterized a variety of parameters including: spatial resolution, uniformity, and distortion using high and low contrast test objects in both horizontal and vertical scanning modes. Test object images depicted hyper- and hypo-echoic masses and demonstrated good resolution, soft tissue contrast and reduced speckle compared to conventional ultrasound images. 
         [0065]    U.S. Pat. No. 4,509,368 to Whiting et al. [26], discloses a method and apparatus for ultrasound tomography for use in clinical diagnostics. The apparatus comprises paired couples of transmission transducers and reflection transducers, independently operable within a container. 
         [0066]    U.S. Pat. No. 7,025,725 to Donald P. Dione et al, [27] discloses an imager having a plurality of cylindrical rings, generating a signal in a cone beam form, U.S. Pat. No. 7,264,592, to Shehada, Ramez E. N. discloses a breast tomography scanner configured to hold fluid within stationary and movable chambers into which a breast is immersed. 
         [0067]    A Lawrence Livermore National Laboratory report [UCRL-JRNL-207220] [16] describes ultrasound tomography using ring geometry for breast imaging. A breast phantom was immersed in a fluid bath. The spatial resolution, deduced from images of reflectivity, was 0.4 mm. The demonstrated 10 cm depth-of-field was superior to that of conventional ultrasound and the image contrast was improved through the reduction of speckle noise and overall lowering of the noise floor. Images of acoustic properties such as sound speed suggested that it is possible to measure variations in the sound speed of 5 m/s. An apparent correlation with X-ray attenuation suggested that the sound speed can be used to discriminate between various types of soft tissue. 
         [0068]    With devices that employ liquid-filled coupling chambers into which the breast is immersed, the transducer is inevitably sited away from the tissue and hence the focus of the ultrasound beam is poor. 
         [0069]    International Patent Publication WO03/103500 [17] discloses a device having a mounting structure capable of holding an ultrasound transducer and a tissue molding element for receiving and surrounding the breast tissue. 
         [0070]    Another system that avoids use of a liquid bath is described by in US Patent Publication 20060241423 of Anderson et al. One side of the breast is compressed by a membrane or film sheet and the other side of the breast is compressed by a rigid plate and an inflatable air bladder. The transducer surface is held against a second surface of the film. An irrigation system is used to maintain a continuous supply of coupling agent at an interface between the transducer surface and the film sheet as the transducer is translated 
         [0071]    Examples of proposed devices for ultrasound-assisted biopsy procedures are disclosed in: U.S. Pat. Nos. 5,660,185 and 5,664,573. A further development and evaluation of a three-dimensional ultrasound-guided breast biopsy apparatus is described in Fenster et al [21]. 
         [0072]    Diffraction tomography (DT) using ultrasound produces a stack of 2D tomography images, similar to those obtained by X-ray or magnetic resonance (MR) tomography. Compared with X-ray or MR DT images, ultrasound CT images do not use potentially harmful ionizing radiation. 
         [0073]    A planar section of a body part being imaged is described by an object function O(r,ω), which for acoustic waves and a lossless object, is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     O 
                      
                     
                       ( 
                       
                         r 
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       k 
                       0 
                       2 
                     
                      
                     
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                             ( 
                             
                               
                                 c 
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         [0000]    where r is the direction of the incident plane wave, c 0  is the speed of sound in the homogeneous background in which the object is immersed, c(r,ω) is the local sound speed inside the object, and k 0  is the background wave number, 2π/λ where λ is the wave length. The goal of DT is to determine the objection function O(r,ω) from a series of diffraction experiments and to generate an image from the object function. 
         [0074]    In ultrasound DT of the breast an array of ultrasound transducers positioned around a ring is used. A breast to be examined is inserted into the ring, and ultrasound images are obtained at each of a sequence of positions of the ring as the ring is moved relative to the breast along an axis essentially perpendicular to the body surface. Each layer of the breast is thus probed using an array of ultrasound transducers arranged along a 360° arc, so that each layer of the breast is probed from essentially all directions. A system for imaging a breast that scans the breast with a circular array of ultrasound transducers is disclosed, for example, in US Patent Publication 2006/0009693[32]. 
         [0075]    DT over an entire circle generates a low-pass-filtered image given by Õ(k)Π(|k|) where Õ(k)=∫ −∞   ∞ d 2 r O(r)e −ik·r  is the two-dimensional Fourier transform of the object function O(r) and 
         [0000]    
       
         
           
             
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         [0000]    where k 0  is the incident wave number. 
         [0076]    DT over an entire circle has the advantage that the breast is probed from all directions. However, due to the anatomy of the breast, the breast can be probed from all directions only for planar sections essentially perpendicular to the axis of the breast. Moreover, because the ring of transducers has a fixed diameter, the distance between the ring and the breast is not uniform as the breast is scanned due to the tapering of the breast towards the nipple. 
         [0077]    Beam forming (BF) methods are an integral part of ultrasound imaging with well known engineering and algorithmic technologies. Consider a body part probed from directions along a limited view circular arc having a central angle 0&lt;ξ&lt;2π. In BF, an object function O(r,ω) is generated by focusing an incident beam from each of a plurality of directions along the circular arc, and for each of these directions of incident acoustic wave, the amplitudes of the scattered rays are determined. The output of this scattering measurement is a set of amplitudes ƒ(φ r ,φ t ), where φ t  is the angle of the incident wave with respect to a fixed radius of the circular arc, and φ r  is the direction of the scattered wave with respect to the fixed radius. The measured ƒ(φ r , φ t ) are phase shifted and integrated over the aperture of the array, so that only the contributions to the scattered field from the focal point are added coherently. As shown by Simonetti and Huang 2008 [31], for a continuous set of transducers, this two-step process is obtained by means of the BF functional 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    where û is the unit vector associated with the angle φ. 
         [0078]    The point spread function associated with the BF functional (2) can be obtained from the free-scattering amplitude defined by: 
         [0000]      ƒ free (φ r ,φ t )= N exp{− ik   0   [û (φ t )− û (φ r )]· r}   (3)
 
         [0000]    where 
         [0000]    
       
         
           
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         [0000]    The beam forming Point Spread function (PSF), also called the Spatial Impulse Response (SIR) is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                                       ) 
                                     
                                   
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                                     ) 
                                   
                                 
                               
                               } 
                             
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                               ( 
                               
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         [0079]    An object O(r) function can be obtained from the relation: I BF (k)=Õ(k) H BF (k), where I BF (k), Õ(k) and H BF (k) are the two-dimensional Fourier transform of the BF functional ℑ BF , O(r), and h BF (z−r), respectively. The image I BF (k) generated from an object function in a beamforming proceedure in which the object is probed from an entire circular arc will in general appear distorted in comparison to an image obtained from a Diffraction Tomography experiment. 
         [0080]    While the “full world” (entire circular arc in a two dimensional world) relationship between beamforming and tomography has been investigated, the extension to a limited-view world (limited-view circular arc in two dimensions and limited-view sphere, such as a hemi-sphere in the three-dimensional world) is not known in the prior art. 
       SUMMARY OF THE INVENTION 
       [0081]    In its first aspect, the present invention provides a system for imaging a body part using ultrasound radiation. 
         [0082]    In one embodiment of the invention, a planar section of the body part is probed using an array of ultrasound transducers that are spatially or temporally arranged on a limited view circular arc having a central angle 0&lt;ξ&lt;2π. In the case of imaging a breast, the use of an arc of transducers allows images of planar sections of the breast to be obtained that are not necessarily perpendicular to the axis of the breast. The beam can be applied onto the breast in a plurality of orientations that are not necessarily perpendicular to the axis of the breast. Thus, planar sections of the breast may be probed sequentially by moving a single array of transducers over the breast, in a variety of directions. 
         [0083]    In this embodiment of the invention, the system includes a processor that is configured to generate an image from a planar section of a body part from the amplitudes ƒ(φ r , φ t ). The inventor has derived an explicit form of the filter g(k) in the relationship 
         [0000]        I   BF ( k )= Õ ( k ) H   BF ( k )= g ( k ) Õ ( k )Π(| k |)  (5)
 
         [0000]    where I BF (k) is the two-dimensional Fourier transform of the limited-view BF functional ℑ BF  in equation (2), H BF (k) is the two-dimensional Fourier transform of h BF (z−r) in equation (4). The derivation of (5) is presented in Annex A. As the term Õ(k)Π(|k|) represents the known result for the full-view diffraction tomography, the filter g(k) constitutes a mapping to obtain the diffraction tomography result. This filter g(k) can be obtained from an explicit form of the limited view H BF (k) (as derived in Annex A): 
         [0000]    
       
         
           
             
               
                 H 
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                           n 
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         [0000]    where I n     1     ,n     2     ,n     2     −n     1    are integrals of products of 3 Bessel Functions of orders n 1 , n 2 , n 2 −n 1 . These integrals are shown in Annex A to include the low pass filter Π(|k|) linearly, i.e H BF (k)=g(k)Π(|k|), thereby defining the explicit form of the filter g(k). 
         [0084]    In accordance with this embodiment of the invention, the processor first calculates I BF (k) as explained above. The I BF (k) is then multiplied by the inverse of the filter (k), to yield Õ(k)Π(|k|), which is the generated tomographic image of the planar section of the body part. Consequently, a tomographic image is generated using beamforming technology. 
         [0085]    In another embodiment of the invention, a three-dimensional section of a body part is probed using an array of ultrasound transducers that are spatially or temporally arranged on a curved surface such as a hemi-sphere. In this embodiment, the scattering amplitudes are given by a function ƒ(θ r , θ t , φ r , φ t ), where (θ t , φ t ) are the spherical coordinates of the transmitted beam and (θ r , φ r ) are the spherical coordinates of the reflected beam, θ r , θ t ε[0,π] and φ r , φ t ε[0,π], (for a shpere φ r , φ t ε[0, 2π]). Standard BF produces the image of an object at a point z, of the image space by focusing an incident beam at r=z in the object space. As in the two dimensional (planar) embodiment, the resulting scattered field is subsequently phase shifted and integrated over the aperture of the array, so that only the contributions to the scattered field from the focal point are added coherently. This two-step process is obtained by means of the 3D BF functional: 
         [0000]    
       
         
           
             
                
               BF 
             
             = 
             
               
                 ∫ 
                 D 
               
                
               
                 
                   exp 
                    
                   
                     [ 
                     
                        
                        
                       
                           
                       
                        
                       
                         k 
                         0 
                       
                        
                       
                         
                           
                             u 
                             ^ 
                           
                            
                           
                             ( 
                             
                               
                                 θ 
                                 r 
                               
                               , 
                               
                                 φ 
                                 r 
                               
                             
                             ) 
                           
                         
                         · 
                         z 
                       
                     
                     ] 
                   
                 
                  
                 
                   f 
                    
                   
                     ( 
                     
                       
                         θ 
                         r 
                       
                       , 
                       
                         θ 
                         t 
                       
                       , 
                       
                         φ 
                         r 
                       
                       , 
                       
                         φ 
                         t 
                       
                     
                     ) 
                   
                 
                  
                 
                   exp 
                    
                   
                     [ 
                     
                        
                        
                       
                           
                       
                        
                       
                         k 
                         0 
                       
                        
                       
                         
                           
                             u 
                             ^ 
                           
                            
                           
                             ( 
                             
                               
                                 θ 
                                 t 
                               
                               , 
                               
                                 φ 
                                 t 
                               
                             
                             ) 
                           
                         
                         · 
                         z 
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
         [0000]    where D is a limited view domain on the sphere. For the special case of a semi-sphere it reads: 
         [0000]      ℑ BF =∫ 0   π   dφ   r ∫ 0   π   dθ   r  sin θ r ∫ 0   π   dφ   t ∫ 0   π   dθ   t  sin θ t ×exp[ ik   0   û (θ r ,φ r )· z ]ƒ(θ r ,θ t ,φ r ,φ t )exp[ ik   0   û (θ t ,φ t )· z]   (6)
 
         [0000]    in the above equations, û is the unit vector associated with the angles θ and φ. As in the two-dimensional embodiment, the second exponential in equation (6) represents focusing in transmission, whereas the first corresponds to the focusing of the received scattered field. The point spread function (PSF) associated with the functional (6) can be obtained by considering the image of a point scatterer at position r. In this case, the three-dimensional free-scattering amplitude is 
         [0000]      ƒ free (θ r ,θ t ,φ r ,φ t )=exp{− ik   0   [û (θ t ,φ t )+ û (θ r ,φ r )]· r}   (7)
 
         [0000]    and the three-dimensional PSF reads 
         [0000]        h   BF =∫ 0   π   dφ   r ∫ 0   π   dθ   r  sin θ r ∫ 0   π   dφ   t ∫ 0   π   dθ   t  sin θ t ×exp{ ik   0   û (θ r ,φ r )·[ z−r ]}exp{ ik   0   û (θ t ,φ t )·[ z−r]}   (8)
 
         [0000]    The inventor developed an analytic expression for the three-dimensional Fourier transform, H BF , of h BF (z−r) in the form: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       BF 
                     
                     = 
                     
                       
                         g 
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                        
                       
                         Π 
                          
                         
                           ( 
                           
                              
                             k 
                              
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       where 
                        
                       
                           
                       
                        
                       
                         Π 
                          
                         
                           ( 
                           
                              
                             k 
                              
                           
                           ) 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             1 
                           
                           
                             
                               
                                  
                                 k 
                                  
                               
                               &lt; 
                               
                                 2 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                  
                                 k 
                                  
                               
                               &gt; 
                               
                                 2 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0086]    The derivation of (9) is presented in Annex B. The DT problem consists of reconstructing the function O(r) from a set of scattering experiments. The object function in the spatial frequency domain, K-space, which is obtained by performing the three-dimensional Fourier transform of O(r) may be represented by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       O 
                       ~ 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                      
                     
                       
                         
                            
                           3 
                         
                          
                         r 
                       
                        
                       
                           
                       
                        
                       
                         O 
                          
                         
                           ( 
                           r 
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                           - 
                           
                              
                              
                             
                                 
                             
                              
                             
                               k 
                               · 
                               r 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The beamforming image: 
         [0000]      ℑ BF =∫ −∞   ∞   dr   1 ∫ −∞   ∞   dr   2 ∫ −∞   ∞   dr   3   O ( r ) h (| z−r |)  (11)
 
         [0000]    in the spatial frequency domain reads (using equation 8) 
         [0000]        I   BF ( k )= Õ ( k ) H   BF ( k )= g ( k ) Õ ( k )Π(| k |)  (12)
 
         [0000]    The derivation of (12) is presented in Annex B. 
         [0087]    While DT over an entire sphere leads to the low-pass-filtered image, Õ(k)Π(|k|), the BF algorithm introduces a distortion that is described by the additional filter g(k) in equation (12). As a result, the DT image can be obtained from the BF image by applying the filter 
         [0000]    
       
         
           
             1 
             
               g 
                
               
                 ( 
                 k 
                 ) 
               
             
           
         
       
     
         [0000]    to the BF image. 
         [0088]    In another of its aspects, the present invention provides an ultrasound apparatus and method for guiding procedures, such as biopsy or ablation in a body part. Real-time three-dimensional images (“4-D ultrasound imaging”) for procedure guidance are superimposed on top of high-resolution tomographic images by spatial registration. This is enabled by mechanically coupling a two-dimensional array transducer for producing three dimensional real-time imaging, to an array of ultrasound transducers arranged on a circular arc of the two dimensional embodiment of the first aspect of the invention or to an array of ultrasound transducers arranged on a hemi-sphere of the three-dimensional embodiment of the first aspect of the invention for producing tomographic imaging. 
         [0089]    Thus, in one of its aspects, the invention provides a system for limited view ultrasound imaging of a 2D section or a 3D volume of a body part comprising: 
         [0000]    (a) one or more ultrasound sensors, the ultrasound sensors being configured to be spatially or temporally arrayed in an array selected from:
 
(i) a limited view circular arc having a central angle ξ, ξ satisfying 0&lt;ξ&lt;2π, the ultrasound sensors generating a plurality of amplitudes ƒ(φ r , φ t ), where ƒ(φ r , φ t ) is an amplitude of ultrasound radiation in a direction forming an angle φ r , with a fixed radius of the limited view circular arc when the body part is probed with incident radiation from a direction forming an angle φ t  with the fixed radius; wherein 0&lt;φ r ,φ t &lt;ξ;
 
(ii) a concave surface, the ultrasound sensors generating a plurality of amplitudes ƒ(θ r , θ t , φ r , φ t ), where ƒ(θ r , θ t , φ r , φ t ) is an amplitude of ultrasound radiation when the body part is probed from a transmit direction determined by angles θ t , φ t  and a receive direction determined by angles θ r , φ r  wherein θ r , θ t ε[0, π] and φ r , φ t ε[0,π];
 
(b) a processor configured to:
 
calculate from the ƒ(φ r , φ t ) or the ƒ(θ r , θ t , φ r , φ t ) a beam forming (BF) functional;
 
calculate free amplitudes ƒ free (φ r , φ t ) or ƒ free (θ r , θ t , φ r , φ t );
 
calculate from the free amplitudes ƒ free (φ r , φ t ) or ƒ free (θ r , θ t , φ r , φ t ) a point spread function (PSF);
 
calculate a filter g(k) from the Fourier transform H BF (k) of the PSF;
 
calculate a Fourier transform I BF (k), of the BF functional;
 
divide I BF (k), by the filter g(k) to yield Õ(k)Π(|k|); and
 
generate an image of 2D section or the 3D volume of the body part using the Õ(k)Π(|k|).
 
         [0090]    The system of the invention may further comprise a scanning device including a dome shaped structure wherein the ultrasound sensors are configured to be spatially or temporally arrayed over at least a portion of the dome structure. The dome shaped structure may be configured to be placed over a breast of a female individual. The dome shaped structure may include a layer formed from an acoustically transparent material. 
         [0091]    The system of the invention may comprise one or more C-Arm-tomography-sensors and one or more 2D-array-sensors. The sensors may be connected to a step-motor-assembly configured to drive the ultrasound sensors over the scanning device. The step-motor-assembly may include any one or more of a motor, an encoder, a processor, an indexer and a driver. The C-arm-tomography-transducer may be moved, for example, along a circular-track. 
         [0092]    The system of the invention may further comprise a display device and the processor may be configured to display the image on the display device. The processor may be further configured to superimpose on a displayed image one or more B-Mode compounded images or tomography images. 
         [0093]    The system of the invention may further comprise a garment configured to be worn by an individual over the body part, the garment comprising a layer formed from a thermo-responsive-acoustic-transparent-polymer, being in a first viscous state at a first temperature below 37° C. and in a second viscous state at a second temperature above 37° C., the second viscous state having a viscosity above a viscosity of the first viscous state. The garment, may be, for example, a bra. 
         [0094]    The system of the invention may further comprise a chair wherein the scanning device is positioned in the chair with the dome in an inverted orientation. 
         [0095]    The thermo-responsive-acoustic-transparent-polymer layer may be harder at an outer surface as compared to an inner surface that is in contact with the body part. 
         [0096]    The dome may comprise one or more holes configured to receive a biopsy needle. 
         [0097]    The invention also provides a garment for use in the system of the invention, the garment being configured to be worn by an individual over the body part, the garment comprising a layer formed from a thermo-responsive-acoustic-transparent-polymer. 
         [0098]    The invention also provides a chair for use in the system of the invention, wherein the scanning device is positioned in the chair with the dome in an adjustable orientation including an inverted orientation. 
         [0099]    The system of the invention may further comprise a 2D array of ultrasound sensors mechanically coupled to a C-arm tomographic arc or to the concave surface and the generated image may be a real-time 3D image. 
         [0100]    In another of its aspects, the invention provides a method for limited view ultrasound imaging of a 2D section or a 3D volume of a body part comprising: 
         [0000]    (a) providing one or more ultrasound sensors, the ultrasound sensors being configured to be spatially or temporally arrayed in an array selected from:
 
(i) a limited view circular arc having a central angle ξ, ξ satisfying 0&lt;ξ&lt;2π, the ultrasound sensors generating a plurality of amplitudes ƒ(φ r , φ t ), where ƒ(φ r , φ t ) is an amplitude of ultrasound radiation in a direction forming an angle φ r  with a fixed radius of the limited view circular arc when the planar section is probed with incident radiation from a direction forming an angle φ t  with the fixed radius; wherein 0&lt;φ r ,φ t &lt;ξ.
 
(ii) a concave surface, the ultrasound sensors generating a plurality of amplitudes ƒ(θ r , θ t , φ r , φ t ), where ƒ(θ r , θ t , φ r , φ t ) is an amplitude of ultrasound radiation when the body part is probed from a transmit direction determined by angles θ t , φ t  and a receive direction determined by angles θ r  φ r  the angles satisfying θ r , θ t ε[0, π] and φ r , φ t ε[0, π],
 
(b) calculating from the ƒ(φ r , φ t ) or the ƒ(θ r , θ t , φ r , φ t ) a beam forming (BF) functional;
 
(c) calculating free amplitudes ƒ free (φ r , φ t ) or the ƒ free (θ r , θ t , φ r , φ t )
 
(d) calculating from the free amplitudes ƒ free (φ r , φ t ) or the ƒ free (θ r , θ t , φ r , φ t ) a point spread function (PSF);
 
(e) calculating a filter g(k) from the Fourier transform H BF (k) of the PSF;
 
(f) calculating a Fourier transform I BF (k), of the BF functional;
 
(g) dividing I BF (k), by the filter g(k) to yield Õ(k)Π(|k|); and
 
(h) generating an image of 2D section or the 3D volume of the body part using the Õ(k)Π(|k|).
 
         [0101]    The body part may be, for example, a breast. 
         [0102]    The method of the invention may further comprise spatially or temporally arraying the ultrasound sensors over at least a portion of the dome structure. The dome shaped structure may include a layer formed from an acoustically transparent material. 
         [0103]    The method of the invention may further comprise displaying the image on a display device. The method may further comprise superimposing on a displayed image one or more B-Mode compounded images or tomography images. 
         [0104]    The method of the invention may further comprise placing a garment on an individual over the body part, the garment comprising a layer formed from a thermo-responsive-acoustic-transparent-polymer, the thermo-responsive-acoustic-transparent-polymer being in a first viscous state at a first temperature below 37° C. and in a second viscous state at a second temperature above 37° C., the second viscous state having a viscosity above a viscosity of the first viscous state. 
         [0105]    In the method of the invention, the scanning device may positioned in a chair with the dome in an adjustable orientation including an inverted orientation, and the method further comprising placing the body part in the dome. A thereto-responsive-acoustic-transparent-polymer that is in a first viscous state at a first temperature below 37° C. and in a second viscous state at a second temperature above 37° C., the second viscous state having a viscosity above a viscosity of the first viscous state may be introduced into the dome. The thermo-responsive-acoustic-transparent-polymer layer may be harder at an outer surface as compared to an inner surface that is in contact with the body part. 
         [0106]    The method of the invention may further comprise inserting a biopsy needle in a hole in the dome structure and obtaining a biopsy. 
         [0107]    The method of the invention may utilize a 2D array of ultrasound sensors that is mechanically coupled to a C-arm tomographic arc or to the concave surface and the method may further comprise generating a real-time 3D image. The real-time 3D image may be used in guiding a surgical or procedure tool through the body part. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0108]    In order to understand the invention and to see how it may be carried out in practice, embodiments will now be described, by way of non-limiting example only, with reference to the accompanying drawings, in which: 
           [0109]      FIG. 1  shows a system for limited view imaging of a body part in accordance with one embodiment of the invention; 
           [0110]      FIG. 2  shows a scanning-device placed on breast for use in the system of  FIG. 1 ; 
           [0111]      FIG. 3  shows internal parts of the scanning device of  FIG. 2 ; 
           [0112]      FIG. 4  shows a cover of the scanning device of  FIG. 2  from a front view ( FIG. 4   a ), a left side view ( FIG. 4   b ), a tilted view ( FIG. 4   c ) and a right side view ( FIG. 4   d ); 
           [0113]      FIG. 5  shows the cup of a bra including a layer of a thermo-responsive acoustic transparent polymeric; 
           [0114]      FIG. 6  shows a C-Arm tomography transducer of the scanning device of  FIG. 2 ; 
           [0115]      FIG. 7  shows a 2D-Array transducer of the scanning device of  FIG. 2 ; 
           [0116]      FIG. 8  shows a C-arm transducer ( FIG. 8   a ) and a 2D-array transducer ( FIG. 8   b ) of the scanning device of  FIG. 2 ; 
           [0117]      FIG. 9  shows schematically the step motor of the scanning device of  FIG. 2 ; 
           [0118]      FIG. 10  shows the C-arm and 2D-array transducers of the scanning device of  FIG. 2  from a top view ( FIG. 10   a ), a diametric view ( FIG. 10   b ), a front view ( FIG. 10   c ) and a right side view ( FIG. 10   d ). 
           [0119]      FIG. 11  shows a chair for use in the system of the invention; 
           [0120]      FIG. 12  shows the scanning device of  FIG. 2 ; and 
           [0121]      FIG. 13  shows the step motor of the scanning device of  FIG. 12 . 
       
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
       [0122]    For the sake of clarity, and ease of description, the present invention will be described in relation to breast imaging, it being evident that the system and method of the invention can be modified to image any desired body part. 
         [0123]      FIG. 1  shows a system  85  for ultrasound imaging of a breast in accordance with one embodiment of the invention. The system  85  comprises a dome shaped scanning device  30 , described in detail below, configured to receive in its interior a breast of an individual  5 . The scanning-device  30  is anchored to an ultrasound system  90  over a cable-assembly  100 . A control-cable  110  connects the ultrasound system  90  to a workstation  120 . The work station  120  may include a CRT screen  123  for displaying images. A user input device, such as a keypad  124  allows a user to input various parameters relating to the examination, such as personal details of the individual being examined, or the parameters of the ultrasound radiation (frequency, intensity, etc.). 
         [0124]    In one embodiment of the invention, shown in  FIG. 1 , the system includes a bra  10  configured to be worn by an individual  5  and the scanning device  30  is configured to be placed on a breast over a cup  20  of the bra. In another embodiment of the invention, shown in  FIG. 11 , the scanning device  30  is incorporated into a chair  7  having a seat  17  upon which the individual  5  is seated. The scanning device  30  is positioned in the chair  7  with its opening on top. The individual  5  sits on the seat  17  and inserts a breast to be imaged into the scanning device  30 . The chair  7  can be aligned in a variety of positions to accommodate individuals of different sizes. 
         [0125]      FIGS. 3 to 10 ,  12  and  13  show the scanning device  30  in greater detail. Referring first to  FIG. 3 , the scanning device includes a dome structure  21  made from an acoustic-transparent-polymer such as Aqualene™. A C-Arm-tomography-transducer  40  and a 2D-Array-transducer  50  are positioned on top of the dome  20 . The transducers  40  and  50  are connected to a step-motor-assembly  80 , which is connected to a scanning-device cover  70  having needle-holes  71 . The cover  70  is shown in greater detail in  FIG. 4 , which shows a front view ( FIG. 4   a ), a left side view ( FIG. 4   b ), a tilted view ( FIG. 4   c ), and a right side view ( FIG. 4   d ) of the cover  70  of-scanning-device connected to the step-motor-assembly  80 . In use, the dome  20  is positioned between the breast and the transducers  40  and  50  which are in contact with the outer-surface  22 . Closer views of the transducers  40  and  50  are shown in  FIGS. 6 and 7 . In  FIG. 6  is shown a concave acoustic-stack  41  of the C-Arm-tomography-transducer  40 . Also shown in  FIG. 6  is a sliding-track  42  of the C-Arm-tomography-transducer  40 .  FIG. 7  shows the acoustic-stack  51  of the 2D-Array-transducer  50  and its sliding-surface  52 . The transducers  40  and  50  are shown with the step-motor-assembly  80  in  FIG. 8  from two directions, a bottom-tilted-view ( FIG. 8   a ) and a top-tilted-view ( FIG. 8   b ). The C-Arm Transducer  40  is connected to the circular-track  71  for enabling rotation by the step-motor-assembly  80 . The acoustic-stacks  41  and  51  are also shown in  FIG. 8   a.    
         [0126]    The step-motor-assembly  80  of the scanning-device  30  is controlled from the workstation  120 ,  FIG. 13  shows the step-motor-assembly  80 , and  FIG. 9  shows schematically the step motor assembly  80 . The motor assembly  80  includes a motor  82  having a rotary axis  81 . An encoder  202  includes a processor  120 , an indexer  84  and a driver  83 . The encoder  202  connects directly to the arc axis of the motor  200  in order to reduce or prevent backlash. In the motor assembly  80 , the gear axis  204  functions as an arc rotation axis. The driver  83  accepts clock pulses and direction signals and translates these signals into appropriate phase currents in the step-motor  82 . The indexer  84  creates the clock pulses and direction signals. The workstation  120  or the processor  121  sends commands to the indexer  84 . 
         [0127]    The step-motor-assembly  80  drives the two transducers  40  and  50  over the dome  20 . In  FIG. 10  is shown a set of views that describe the direction of motion driven by the step-motor-assembly  80 . A view from above is shown in  FIG. 10   a , a dimeric view is shown in  FIG. 10   b  a front view is shown in  FIG. 10   c , and a right side view is shown in  FIG. 10   d . The rotate-arrow  85  shows the direction of rotation of the C-Arm-tomography-transducer  40  along the circular-track  71  ( FIG. 8 ), the tilt-rotation-arrow  87  shows the tilting rotation of the C-Arm-tomography-transducer  40  and the slide-double-arrow  86  shows the direction the 2D-Array-transducer  50  slides along the C-Arm-tomography-transducer  40 . The motions indicated by the arrows  85 ,  86  and  87  are all driven by the step-motor-assembly  80 . 
         [0128]      FIG. 12  shows the scanning device and step motor in greater detail. For each planar section of the breast to be imaged, the transducers  40  and  50  are moved along circular arcs. The adaptors  61  are made of an acoustically transparent material such as Aqualene™ to assure that there is no air in between the acoustic stack of the transducers and the dome  20 . The plane of the section is not necessarily perpendicular to the axis of the breast. The orientation of the circular arc is monitored by the step motor  200  and is continuously input to the processor  121 . The transducers  40  and  50  may act as B-mode ultrasound probes, enabling compound imaging of the images obtained from these transducers. Alternatively, for each pair of a receive transducer and a transmit transducer transmission signals may be measured. The transmission images may be combined with the B-Mode compounded images or with the reflection tomography image produced by an arc of piesoelectric sensors. 
         [0129]      FIG. 2  shows the individual with the scanning device  30  placed on a cup  20  of the bra  10  shown in greater detail in  FIG. 5 . The cup  20  includes an outer fabric layer  23  and an inner fabric layer  25 . Between the inner and outer layers is a thereto-responsive-acoustic-transparent-polymer  27 . The state of the thermo-responsive acoustic-transparent polymer  27  is temperature dependent, so that at room temperature it is in a liquid state, while at body temperature (˜37° C.) it is in a solid state. An example such a polymer is the nonionic surfactant polyol, copolymer poloxamer  407  also known as Pluronic F127™. A discussion on the safety of polyol, copolymer poloxamer  407  when in contact with human tissue can be found in Khattak et al [49]. 
         [0130]    The breast is inserted into the dome  21  with the polymeric material in a viscous form, so that the inner surface of the polymeric material conforms to the shape of the breast surface before solidifying in the shape of the breast surface. The polymeric material may be harder at its outer surface that is in contact with the acoustic stack, as compared to the inner surface that is in contact with the anatomy of the breast. This gradient of hardness along the polymeric material enables producing a perfect outer spherical surface, while keeping the flexibility of adjusting to the complicated surface of the breast. Alternatively, the overlying dome forces the outer surface of the polymeric material to adopt a spherical shape. The polymeric material is an acoustically coupling material which acoustically couples the breast surface with the transducers on the outer surface of the dome. This enables the inner surface of the cup  20  of the bra to conform to the surface of the breast, so that no air is present between the cup and the breast. This allows scanning of the breast with the breast in its natural shape. The thermo-responsive acoustic-transparent polymer  20  may be sterilizable. 
         [0131]    In use, breast is inserted into the dome  21  of the scanning device  30 . If the bra  10  is being worn, the thermo-responsive-acoustic-transparent-polymer may also be introduced between the bra and the inner surface of the dome  20 , so that no air is present between the outer surface of the bra and the inner surface of the dome. Alternatively, if the chair  7  is being used ( FIG. 11 ), then the inverted dome  30  may be filled with the thereto-responsive-acoustic-transparent-polymer before insertion of the breast. 
         [0132]    After application of the scanning device  30  to the breast, the transducers  40  and  50  are driven one at a time, and for each driven transducer, each transducer detects ultrasound radiation. The ultrasound wave detected by each transducer is converted by the transducer into an electric signal indicative of the amplitude of the detected wave (ƒ(φ r , φ t ) in the case of 2D tomography or ƒ(θ r , θ t , φ r , φ t ) in the case of 3D tomography) that is input to the ultrasound system  90  via the cable  100 . 
         [0133]    The ultrasound system  90  includes a processor configured to generate a 2D or a 3D image from the signals input from the transducers. As explained above, I BF (k) is first calculated. The I DT (k) is then calculated by multiplying by the inverse of the filter g(k) to yield I DT (k)=Õ(k)Π(|k|), which a tomographic image of the breast. This tomographic image may be combined with the compounded images of B-Mode and with the transmission mode images from the transducers  40  and  50 . Superimposing these separate types of images is possible due to the fact that these alternative hardware configurations are mechanically coupled to the arc, so that spatial registration is possible. 
         [0134]    The 2-D acoustic stack array  51  produces real-time 3-D images (“4-D ultrasound imaging”) for procedure guidance, such as guiding a needle in biopsy or guiding ablation devices. The sliding track  42  and the sliding surface  52  are used for placing the 2-D transducer array in an optimal location with respect to the breast for the guidance procedure. When the 2-D transducer array  50  is operated, the C-arm tomography transducer  40  is kept static. The procedure device, such as the needle  60  in  FIG. 3  can be inserted through the needle holes in the cover  70  and through the thermo-responsive acoustic-transparent polymer  20 . Mechanical attachment of the 2-D transducer array  50  to the C-arm tomography transducer  40  allows superposition of the real-time 3D images on top of high-resolution tomographic images produced by the C-arm tomography transducer  40 . 
       ANNEX A 
     Difraction Tomography Algorithm for the Limited View Concave Aperture 
       [0135]    A new derivation of a two-dimensional DT based on a two-dimensional beamforming (BF) algorithm is discussed, as an alternative approach to standard DT algorithms such as the filtered backpropagation method 1 .  1  Devaney, A. J. 1982, “A filtered backpropagation algorithm for diffraction tomography”, Ultrason. Imaging 4, 336-350. 
         [0136]    We assume that the scattering problem is described by a scalar wave field, ψ, solution to 
         [0000]        H ψ( r,k   0   ,{circumflex over (r)}   0 ,ω)=− O ( r ,ω)ψ( r,k   0   {circumflex over (r)}   0 ,ω)  (13)
 
         [0000]    where H is the Helmholtz operator in 2D 
         [0000]    
       
         
           
             
               ( 
               
                 
                   
                     ∂ 
                     2 
                   
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 + 
                 
                   
                     ∂ 
                     2 
                   
                   
                     ∂ 
                     
                       y 
                       2 
                     
                   
                 
                 + 
                 
                   k 
                   0 
                   2 
                 
               
               ) 
             
             , 
           
         
       
     
         [0000]    k 0  is the background wavenumber (2π/λ), {circumflex over (r)} 0  specifies the direction of an incident plane wave that illuminates the object and ω is the angular frequency. The unit vector {circumflex over (r)} 0  is defined by the polar angle φ t . 
         [0137]    The object is described by the so-called Object Function that depends on the type of wave field used to probe the object: for electromagnetic wave sensing, it is related to the index of refraction 2 , n(r,ω), through the relation O(r)=k 0   2 [n 2 (r,ω)−1], and for acoustic waves, it is linked to the speed of sound and the attenuation coefficient 3 . In particular, for a lossless object 
         [0000]    
       
         
           
             
               
                 
                   
                     O 
                      
                     
                       ( 
                       
                         r 
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       k 
                       0 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 c 
                                 0 
                               
                               
                                 c 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     , 
                                     ω 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         1 
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where c 0  is the sound speed of the homogeneous background in which the object is immersed and c(r,ω) is the local sound speed inside the object. The dependence of the object function on ω is because of dispersion and energy dissipation phenomena. The analysis performed in the rest of this section will consider monochromatic wave fields; therefore, the explicit dependence on ω is omitted.  2  Born, M. &amp; Wolf, E. 1999 Principles of optics. Cambridge, UK: Cambridge University Press. 3  Kak, A. C. &amp; Slaney, M. 1988 Principles of computerized tomographic imaging. New York, N.Y.: IEEE Press. 
       Two-Dimensional Beamforming Algorithm Over a Limited View Arc 
       [0138]    Let us assume that the scattering amplitude, ƒ(φ r , φ t ), can be measured as a continuous function of the illumination and detection directions, φ r , φ t ε[0,ξ], (note that for a full circle φ r , φ t ε[0,2π]), these angles corresponding to the angles relative to the x-axis of a standard polar coordinate system. In principle, this could be achieved with the array of transreceivers that partially surrounds the object, placed on a limited view circular arc. 
         [0139]    Standard BF produces the image of an object at a point, z, of the image space by focusing an incident beam at r=z in the object space. The resulting scattered field is subsequently phase shifted and integrated over the aperture of the array, so that only the contributions to the scattered field from the focal point are added coherently. This two-step process is obtained by means of the BF functional 
         [0000]      ℑ BF =∫ 0   ξ   dφ   r ∫ 0   ξ   dφ   t ×exp[− ik   0   û (φ r )· z ]ƒ(φ r ,φ t )exp[ ik   0   û (φ t )· z]   (15)
 
         [0000]    where û is the unit vector associated with the angle φ. As discussed by 4  for the full circle two-dimensional case, the second exponential in equation (III) represents focusing in transmission, whereas the first corresponds to the focusing of the received scattered field. The point spread function (PSF) associated with the functional (2) can be obtained by considering the image of a point scatterer at position r. In this case, the free-scattering amplitude is 
         [0000]      ƒ free (φ r ,φ t )= N exp{− ik   0   [û (φ t )− û (φ r )]· r}   (16)
 
         [0000]    where 
         [0000]    
       
         
           
             
               = 
               
                 
                   exp 
                    
                   
                     ( 
                     
                       
                          
                          
                         
                             
                         
                          
                         π 
                       
                       4 
                     
                     ) 
                   
                 
                 
                   
                     8 
                      
                     π 
                      
                     
                         
                     
                      
                     
                       k 
                       0 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    and the Point Spread function (PSF), also called the Spatial Impulse Response (SIR) reads 
         [0000]        h   BF ( z−r )= N∫   0   ξ   dφ   r ∫ 0   ξ   dφ   t ×exp{− ik   0   û (φ r )·( z−r )}exp{ ik   0   û (φ t )·( z−r )}  (17)
 
         [0000]    û(φ r )·(z−r)=|z−r| cos α, where α is the angle between the receive unit vector û(φ r ) and the vector z−r. Denoting the angle of z−r as φ′, α=φ r −φ′.  4  Simonetti, F. &amp; Huang, L. 2008, “From beamforming to diffraction tomography”, J. Appl. Phys. 103, 103 110. 
         [0140]    The Jacobi-Anger expansion gives: 
         [0000]      exp{ ik   0   |z−r |cos(α)}= J   0 ( k   0   |z−r |)+2Σ n=1   ∞   i   n   J   n ( k   0   |z−r |)cos( n α)
 
         [0000]    where J n  is the Bessel function of the order n. 
         [0141]    We have 5 : 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 ξ 
               
                
               
                 
                    
                   
                     φ 
                     r 
                   
                 
                  
                 exp 
                  
                 
                   { 
                   
                      
                      
                     
                         
                     
                      
                     
                       k 
                       0 
                     
                      
                     
                       
                         
                           u 
                           ^ 
                         
                          
                         
                           ( 
                           
                             φ 
                             r 
                           
                           ) 
                         
                       
                       · 
                       
                         ( 
                         
                           z 
                           - 
                           r 
                         
                         ) 
                       
                     
                   
                   } 
                 
               
             
             = 
             
               
                 
                   ξ 
                    
                   
                       
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   2 
                    
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                       
                          
                         n 
                       
                        
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           ∫ 
                           0 
                           ξ 
                         
                          
                         
                           
                              
                             
                               φ 
                               r 
                             
                           
                            
                           
                             cos 
                              
                             
                               [ 
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     
                                       φ 
                                       r 
                                     
                                     - 
                                     
                                       φ 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               == 
               
                 
                   ξ 
                    
                   
                       
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   2 
                    
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                       
                          
                         n 
                       
                        
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           ∫ 
                           0 
                           ξ 
                         
                          
                         
                            
                           
                             
                               φ 
                               r 
                             
                              
                             
                               [ 
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         
                                           φ 
                                           r 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     ∫ 
                     0 
                     ξ 
                   
                    
                   
                     
                        
                       
                         φ 
                         r 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         
                           n 
                            
                           
                               
                           
                            
                           
                             φ 
                             r 
                           
                         
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       
                         1 
                         n 
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             n 
                              
                             
                                 
                             
                              
                             
                               φ 
                               r 
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       | 
                       0 
                       ξ 
                     
                   
                   = 
                   
                     
                       1 
                       n 
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           n 
                            
                           
                               
                           
                            
                           ξ 
                         
                         ) 
                       
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
               
                 ∫ 
                 0 
                 ξ 
               
                
               
                 
                    
                   
                     φ 
                     r 
                   
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     
                       n 
                        
                       
                           
                       
                        
                       
                         φ 
                         r 
                       
                     
                     ) 
                   
                 
               
             
             = 
             
               
                 
                   
                     - 
                     
                       1 
                       n 
                     
                   
                    
                   
                     cos 
                      
                     
                       ( 
                       
                         n 
                          
                         
                             
                         
                          
                         
                           φ 
                           r 
                         
                       
                       ) 
                     
                   
                 
                  
                 
                   | 
                   0 
                   ξ 
                 
               
               = 
               
                 
                   - 
                   
                     
                       1 
                       n 
                     
                      
                     
                       [ 
                       
                         
                           cos 
                            
                           
                             ( 
                             
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                         - 
                         1 
                       
                       ] 
                     
                   
                 
                 = 
                 
                   
                     1 
                     n 
                   
                    
                   
                     [ 
                     
                       1 
                       - 
                       
                         cos 
                          
                         
                           ( 
                           
                             n 
                              
                             
                                 
                             
                              
                             ξ 
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ∫ 
                 0 
                 ξ 
               
                
               
                 
                    
                   
                     φ 
                     r 
                   
                 
                  
                 exp 
                  
                 
                   { 
                   
                      
                      
                     
                         
                     
                      
                     
                       k 
                       0 
                     
                      
                     
                       
                         
                           u 
                           ^ 
                         
                          
                         
                           ( 
                           
                             φ 
                             r 
                           
                           ) 
                         
                       
                       · 
                       
                         [ 
                         
                           z 
                           - 
                           r 
                         
                         ] 
                       
                     
                   
                   } 
                 
               
             
             == 
             
               
                 ξ 
                  
                 
                     
                 
                  
                 
                   
                     J 
                     0 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                     ) 
                   
                 
               
               + 
               
                 2 
                  
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       
                          
                         n 
                       
                       n 
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             sin 
                              
                             
                               ( 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 
                                   φ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             [ 
                             
                               1 
                               - 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 
                                   φ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
           
         
       
     
         [0000]      5  sin(A+B)=sin A cos B+cos A sin B; sin(A−B)=sin A cos B−cos A sin B; cos(A+B)=cos A cos B−sin A sin B; cos(A−B)=cos A cos B+sin A sin B http://www.ies.co.jp/math/java/trig/kahote/kahote.html 
         [0142]    The complex conjugate result is obtained for the transmit angles. h BF (z−r) is therefore given by 
         [0000]    
       
         
           
             
               
                 h 
                 BF 
               
                
               
                 ( 
                 
                   z 
                   - 
                   r 
                 
                 ) 
               
             
             = 
             
               
                 
                   N 
                   ( 
                   
                     
                       ξ 
                        
                       
                           
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       2 
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                           
                             
                               i 
                               n 
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             n 
                                              
                                             
                                                 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             n 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               φ 
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         1 
                                         - 
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               n 
                                                
                                               
                                                   
                                               
                                                
                                               ξ 
                                             
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           n 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   ) 
                 
                 * 
               
               × 
               
                 ( 
                 
                   
                     ξ 
                      
                     
                         
                     
                      
                     
                       
                         J 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     2 
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             i 
                             n 
                           
                           n 
                         
                          
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                       ) 
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           n 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             n 
                                              
                                             
                                                 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0143]    Special Cases: 
         [0144]    Note that when ξ=2π (i.e. a full-circle), the received and transmitted beams read: 
         [0000]      ∫ 0   2π   dφ   r exp{ ik   0   û (φ r )·( z−r )}=2 πJ   0 ( k   0   |z−r |)
 
         [0145]    So h BF (|z−r|)=4π 2 NJ 0   2 (k 0 |z−r|), for ξ=2π 
         [0146]    Note also that for ξ=π (i.e. a semi-circle) and φ′=0, or φ′=multiples of π (i.e. a focal point and field point along [or parallel to] the x axis): 
         [0000]      ∫ 0   π   dφ   r exp{ ik   0   û (φ r )·( z−r )}=π J   0 ( k   0   |z−r |),φ′=0,or φ′=multiples of π
 
         [0147]    So h BF (|z−r|)=π 2 NJ 0   2 (k 0 |z−r|), for ξ=π, and φ′=0, or φ′=multiples of π 
         [0148]    We now calculate the two-dimensional Fourier transform H BF (k) of h BF (z−r): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         H 
                         BF 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                       
                      
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                              
                             2 
                           
                            
                           
                             
                               rh 
                               BF 
                             
                              
                             
                               ( 
                               
                                 z 
                                 - 
                                 r 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               k 
                               · 
                               
                                 [ 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                              
                             2 
                           
                            
                           r 
                         
                          
                         
                             
                         
                          
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               k 
                               · 
                               
                                 [ 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ] 
                               
                             
                           
                         
                         × 
                       
                     
                   
                 
               
               
                 
                   
                       
                      
                     
                       N 
                       ( 
                       
                         
                           ξ 
                            
                           
                               
                           
                            
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                         + 
                       
                     
                   
                 
               
               
                 
                   
                       
                      
                     
                       2 
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                           
                             
                               i 
                               n 
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             { 
                             
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                  
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         
                             
                            
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   
                                     φ 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                           
                           } 
                         
                         ) 
                       
                       * 
                     
                     × 
                   
                 
               
               
                 
                   
                       
                      
                     
                       ( 
                       
                         
                           ξ 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                         + 
                       
                     
                   
                 
               
               
                 
                   
                       
                      
                     
                       2 
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                           
                             
                               i 
                               n 
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             { 
                             
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                  
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                         
                        
                       
                         
                           [ 
                           
                             1 
                             - 
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   ξ 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               n 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                             ) 
                           
                         
                       
                       } 
                     
                     ) 
                   
                 
               
             
               
           
         
       
     
         [0149]    Denoting the angle of k as φ, α=φ′−φ; k·[z−r]=|k∥z−r|cos(α), we now use the Jacobi-Anger expansion again with: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   
                     
                       exp 
                        
                       
                         { 
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             k 
                             · 
                             
                               [ 
                               
                                 z 
                                 - 
                                 r 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                     = 
                       
                      
                     
                       exp 
                        
                       
                         { 
                         
                           
                             - 
                              
                           
                            
                           
                              
                             k 
                              
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         [ 
                         
                           
                             
                               J 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             2 
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   i 
                                   n 
                                 
                                  
                                 
                                   
                                     J 
                                     n 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                          
                                         k 
                                          
                                       
                                        
                                       
                                          
                                         
                                           z 
                                           - 
                                           r 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       α 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       * 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 H 
                 BF 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 r 
                  
                 
                     
                 
                  
                 
                    
                   r 
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         
                           
                             φ 
                             ′ 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   J 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                        
                                       k 
                                        
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 2 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       
                                         n 
                                         3 
                                       
                                       = 
                                       1 
                                     
                                     ∞ 
                                   
                                    
                                   
                                     
                                       i 
                                       
                                         n 
                                         3 
                                       
                                     
                                      
                                     
                                       
                                         
                                           J 
                                           
                                             n 
                                             3 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                                
                                               k 
                                                
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         3 
                                                       
                                                        
                                                       φ 
                                                     
                                                     ) 
                                                   
                                                    
                                                   
                                                     cos 
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           n 
                                                           3 
                                                         
                                                          
                                                         
                                                           φ 
                                                           ′ 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 + 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 sin 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       n 
                                                       3 
                                                     
                                                      
                                                     φ 
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   sin 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         3 
                                                       
                                                        
                                                       
                                                         φ 
                                                         ′ 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         * 
                       
                     
                     × 
                     × 
                     
                       
                         N 
                          
                         
                           ( 
                           
                             
                               ξ 
                                
                               
                                   
                               
                                
                               
                                 
                                   J 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               2 
                                
                               
                                 
                                   ∑ 
                                   
                                     
                                       n 
                                       2 
                                     
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     
                                       i 
                                       
                                         n 
                                         2 
                                       
                                     
                                     
                                       n 
                                       2 
                                     
                                   
                                    
                                   
                                     
                                       J 
                                       
                                         n 
                                         2 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           0 
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     { 
                                     
                                       
                                         
                                           
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     n 
                                                     2 
                                                   
                                                    
                                                   ξ 
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 cos 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       n 
                                                       2 
                                                     
                                                      
                                                     
                                                       φ 
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               [ 
                                               
                                                 1 
                                                 - 
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         2 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                              
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     n 
                                                     2 
                                                   
                                                    
                                                   
                                                     φ 
                                                     ′ 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       * 
                     
                     × 
                     
                       ( 
                       
                         
                           ξ 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           2 
                            
                           
                             
                               ∑ 
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 1 
                               
                               ∞ 
                             
                              
                             
                               
                                 
                                   i 
                                   
                                     n 
                                     1 
                                   
                                 
                                 
                                   n 
                                   1 
                                 
                               
                                
                               
                                 
                                   J 
                                   
                                     n 
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 { 
                                 
                                   
                                     
                                       
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                                
                                               ξ 
                                             
                                             ) 
                                           
                                            
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                  
                                                 
                                                   φ 
                                                   ′ 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           [ 
                                           
                                             1 
                                             - 
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     n 
                                                     1 
                                                   
                                                    
                                                   ξ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                          
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                                
                                               
                                                 φ 
                                                 ′ 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 } 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0150]    Integration of the angle φ′ is over simple products of sin and cos trigonometric functions and can be worked out easily. 
         [0151]    We therefore, now focus on the integral of a product of 3 Bessel functions. Such integrals are available in closed form in the literature 6 . For example: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                   t 
                   
                     1 
                     - 
                     
                       n 
                       1 
                     
                   
                 
                  
                 
                   
                     J 
                     
                       n 
                       1 
                     
                   
                    
                   
                     ( 
                     at 
                     ) 
                   
                 
                  
                 
                   
                     J 
                     
                       n 
                       2 
                     
                   
                    
                   
                     ( 
                     bt 
                     ) 
                   
                 
                  
                 
                   
                     J 
                     
                       n 
                       2 
                     
                   
                    
                   
                     ( 
                     ct 
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                    
                   t 
                 
               
             
             = 
             
               
                 
                   
                     
                       ( 
                       bc 
                       ) 
                     
                     
                       
                         n 
                         1 
                       
                       - 
                       1 
                     
                   
                    
                   
                     
                       sin 
                       
                         
                           n 
                           1 
                         
                         - 
                         
                           1 
                           2 
                         
                       
                     
                      
                     
                       ( 
                       A 
                       ) 
                     
                   
                 
                 
                   
                     
                       ( 
                       
                         2 
                          
                         π 
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                    
                   
                     a 
                     
                       n 
                       1 
                     
                   
                 
               
                
               
                 
                   P 
                   
                     
                       n 
                       2 
                     
                     - 
                     
                       1 
                       2 
                     
                   
                   
                     
                       1 
                       2 
                     
                     - 
                     
                       n 
                       1 
                     
                   
                 
                  
                 
                   ( 
                   
                     cos 
                      
                     
                         
                     
                      
                     A 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   R 
                    
                   
                     ( 
                     
                       n 
                       1 
                     
                     ) 
                   
                 
                 &gt; 
                 
                   - 
                   
                     1 
                     2 
                   
                 
               
               , 
               
                 
                   R 
                    
                   
                     ( 
                     
                       n 
                       2 
                     
                     ) 
                   
                 
                 &gt; 
                 
                   - 
                   
                     1 
                     2 
                   
                 
               
             
           
         
       
     
         [0000]      6  Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962, p. 331 and 332 
         [0152]    If a, b, c are sides of a triangle of area Δ. And A is 
         [0000]    
       
         
           
             A 
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             0 
                             , 
                           
                         
                         
                           
                             
                               a 
                               2 
                             
                             &lt; 
                             
                               
                                 
                                   ( 
                                   
                                     b 
                                     - 
                                     c 
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                   
                               
                                
                               and 
                                
                               
                                   
                               
                                
                               
                                 a 
                                 2 
                               
                             
                             &lt; 
                             
                               
                                 ( 
                                 
                                   b 
                                   + 
                                   c 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       
                         
                           
                             
                               arc 
                                
                               
                                   
                               
                                
                               cos 
                                
                               
                                 
                                   
                                     b 
                                     2 
                                   
                                   + 
                                   
                                     c 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     2 
                                   
                                 
                                 
                                   2 
                                    
                                   bc 
                                 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   b 
                                   - 
                                   c 
                                 
                                 ) 
                               
                               2 
                             
                             ≤ 
                             
                               a 
                               2 
                             
                             ≤ 
                             
                               
                                 ( 
                                 
                                   b 
                                   + 
                                   c 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       
                         
                           
                             π 
                             , 
                           
                         
                         
                           
                             
                               a 
                               2 
                             
                             &gt; 
                             
                               
                                 
                                   ( 
                                   
                                     b 
                                     - 
                                     c 
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                   
                               
                                
                               and 
                                
                               
                                   
                               
                                
                               
                                 a 
                                 2 
                               
                             
                             &gt; 
                             
                               
                                 ( 
                                 
                                   b 
                                   + 
                                   c 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     Δ 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     bc 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         A 
                         ) 
                       
                     
                   
                 
                 , 
                 
                   
                     or 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         A 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       2 
                        
                       Δ 
                     
                     bc 
                   
                 
               
             
           
         
       
     
         [0153]    P υ   μ  are legendre functions 7  of the first kind: 
         [0000]    
       
         
           
             
               
                 P 
                 λ 
                 μ 
               
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 
                   
                     1 
                     
                       Γ 
                        
                       
                         ( 
                         
                           1 
                           - 
                           μ 
                         
                         ) 
                       
                     
                   
                    
                   
                     [ 
                     
                       
                         1 
                         + 
                         z 
                       
                       
                         1 
                         - 
                         z 
                       
                     
                     ] 
                   
                 
                 
                   μ 
                   2 
                 
               
                
               
                 
                   F 
                   1 
                     
                     
                   2 
                     
                 
                  
                 
                   ( 
                   
                     
                       - 
                       λ 
                     
                     , 
                     
                       λ 
                       + 
                       1 
                     
                     , 
                     
                       1 
                       - 
                       μ 
                     
                     , 
                     
                       
                         1 
                         - 
                         z 
                       
                       2 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   F 
                   1 
                     
                     
                   2 
                     
                 
                  
                 
                   ( 
                   
                     a 
                     , 
                     b 
                     , 
                     c 
                     , 
                     z 
                   
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     0 
                   
                   ∞ 
                 
                  
                 
                   
                     
                       
                         
                           ( 
                           a 
                           ) 
                         
                         n 
                       
                        
                       
                         
                           ( 
                           b 
                           ) 
                         
                         n 
                       
                     
                     
                       
                         ( 
                         c 
                         ) 
                       
                       n 
                     
                   
                    
                   
                     
                       z 
                       n 
                     
                     
                       n 
                       ! 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               provided 
                
               
                   
               
                
               that 
                
               
                   
               
                
               c 
                
               
                   
               
                
               is 
                
               
                   
               
                
               not 
                
               
                   
               
                
               0 
             
             , 
             
               - 
               1 
             
             , 
             
               - 
               2 
             
             , 
             … 
              
             
                 
             
             , 
             
               
                 
                   and 
                    
                   
                     
 
                   
                   ( 
                   a 
                   ) 
                 
                 n 
               
               = 
               
                 
                   a 
                    
                   
                     ( 
                     
                       a 
                       + 
                       1 
                     
                     ) 
                   
                 
                  
                 
                   ( 
                   
                     a 
                     + 
                     2 
                   
                   ) 
                 
                  
                 
                     
                 
                  
                 … 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     a 
                     + 
                     n 
                     - 
                     1 
                   
                   ) 
                 
               
             
             , 
             
               
                 
                   ( 
                   a 
                   ) 
                 
                 0 
               
               = 
               1 
             
           
         
       
     
         [0000]      7  http://en.wikipedia.org/wiki/Legendre_function; http://en.wikipedia.org/wiki/Hypergeometric_function http://en.wikipedia.org/wiki/Gamma_function
 
provided that c is not 0, −1, −2, . . . , and (a) n =a(a+1)(a+2) . . . (a+n−1), (a) 0 =1
 
         [0154]    For the particular case of the same index of all three Bessel functions: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           t 
                           
                             1 
                             - 
                             n 
                           
                         
                          
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             at 
                             ) 
                           
                         
                          
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             bt 
                             ) 
                           
                         
                          
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             ct 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           t 
                         
                       
                     
                     = 
                     
                       
                         
                           2 
                           
                             n 
                             - 
                             1 
                           
                         
                          
                         
                           Δ 
                           
                             
                               2 
                                
                               n 
                             
                             - 
                             1 
                           
                         
                       
                       
                         
                           
                             π 
                              
                             
                               ( 
                               abc 
                               ) 
                             
                           
                           n 
                         
                          
                         
                           
                             ( 
                             
                               1 
                               2 
                             
                             ) 
                           
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   
                     = 
                     
                       &gt; 
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                          
                         
                           t 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               at 
                               ) 
                             
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               bt 
                               ) 
                             
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               ct 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             t 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     = 
                     
                       1 
                       
                         2 
                          
                         πΔ 
                       
                     
                   
                 
               
             
               
           
         
       
     
         [0155]    For our case a=b=k 0 , c=|k| we get: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             A 
                           
                           = 
                           
                             
                               
                                 b 
                                 2 
                               
                               + 
                               
                                 c 
                                 2 
                               
                               - 
                               
                                 a 
                                 2 
                               
                             
                             
                               2 
                                
                               bc 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
                                 k 
                                 0 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   k 
                                    
                                 
                                 2 
                               
                               - 
                               
                                 k 
                                 0 
                                 2 
                               
                             
                             
                               2 
                                
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 k 
                                  
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                                
                               k 
                                
                             
                             
                               2 
                                
                               
                                 k 
                                 0 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       sin 
                        
                       
                         ( 
                         A 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           
                             
                               cos 
                               2 
                             
                              
                             A 
                           
                         
                       
                       = 
                       
                         
                           
                             1 
                             - 
                             
                               
                                 
                                    
                                   k 
                                    
                                 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   k 
                                   0 
                                   2 
                                 
                               
                             
                           
                         
                         . 
                         
                           
 
                         
                          
                         Therefore 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     Δ 
                     = 
                     
                       
                         
                           1 
                           2 
                         
                          
                         bc 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             A 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           1 
                           2 
                         
                          
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         
                           
                             1 
                             - 
                             
                               
                                 
                                    
                                   k 
                                    
                                 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   k 
                                   0 
                                   2 
                                 
                               
                             
                           
                         
                          
                         
                             
                         
                          
                         and 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         t 
                          
                         
                             
                         
                          
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             at 
                             ) 
                           
                         
                          
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             bt 
                             ) 
                           
                         
                          
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             ct 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           t 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           2 
                            
                           πΔ 
                         
                       
                       = 
                       
                         1 
                         
                           π 
                            
                           
                               
                           
                            
                           
                             k 
                             0 
                           
                            
                           
                              
                             k 
                              
                           
                            
                           
                             
                               1 
                               - 
                               
                                 
                                   
                                      
                                     k 
                                      
                                   
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     k 
                                     0 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0156]    We therefore arrived at a formula for H BF  of the limited view arc. The triangular relation for a, b, c in our case requires that |k|≦2k 0  so we get the low pass filtering: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       BF 
                     
                     = 
                     
                       
                         g 
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                        
                       
                         Π 
                          
                         
                           ( 
                           
                              
                             k 
                              
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       where 
                        
                       
                           
                       
                        
                       
                         Π 
                          
                         
                           ( 
                           
                              
                             k 
                              
                           
                           ) 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             1 
                           
                           
                             
                               
                                  
                                 k 
                                  
                               
                               &lt; 
                               
                                 2 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                  
                                 k 
                                  
                               
                               &gt; 
                               
                                 2 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0157]    The DT problem consists of reconstructing the function O(r) from a set of scattering experiments. For this purpose, it is convenient to introduce the representation of the object function in the spatial frequency domain, K-space, which is obtained by performing the two-dimensional Fourier transform of O(r) 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       O 
                       ~ 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                            
                           2 
                         
                          
                         r 
                       
                        
                       
                           
                       
                        
                       
                         O 
                          
                         
                           ( 
                           r 
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             k 
                             · 
                             r 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0158]    We now look at the beamforming image: 
         [0000]      ℑ BF =∫ −∞   ∞   dr   1 ∫ −∞   ∞   dr   2   O ( r ) h (| z−r |)  (21)
 
         [0000]    that in the spatial frequency domain reads 
         [0000]        I   BF ( k )= Õ ( k ) H   BF ( k )= g ( k ) Õ ( k )Π(| k |)  (22)
 
         [0159]    While DT over the entire circle leads to the low-pass-filtered image, Õ(k)Π(|k|), the new BF algorithm introduces a distortion that is described by the additional filter g(k). As a result, the DT image can be obtained from the BF image by applying the filter 
         [0000]    
       
         
           
             1 
             
               g 
                
               
                 ( 
                 k 
                 ) 
               
             
           
         
       
     
         [0000]    to the BF image. Again, this is an alternative approach to other DT algorithms 8 .  8  Reference of footnote 3 
       ANNEX AI 
     Integrals 
       [0160]    n 1 =0, n 2  for the general formula: 
         [0000]    
       
         
           
             
               
                 
                   
                     ∫ 
                     0 
                     ∞ 
                   
                    
                   
                     
                       t 
                       
                         1 
                         - 
                         
                           n 
                           1 
                         
                       
                     
                      
                     
                       J 
                       
                         n 
                         1 
                       
                     
                      
                     at 
                   
                 
                 ) 
               
                
               
                 
                   J 
                   
                     n 
                     2 
                   
                 
                  
                 
                   ( 
                   bt 
                   ) 
                 
               
                
               
                 
                   J 
                   
                     n 
                     2 
                   
                 
                  
                 
                   ( 
                   ct 
                   ) 
                 
               
                
               
                   
               
                
               
                  
                 t 
               
             
             = 
             
               
                 
                   
                     
                       ( 
                       bc 
                       ) 
                     
                     
                       
                         n 
                         1 
                       
                       - 
                       1 
                     
                   
                    
                   
                     
                       sin 
                       
                         
                           n 
                           1 
                         
                         - 
                         
                           1 
                           2 
                         
                       
                     
                      
                     
                       ( 
                       A 
                       ) 
                     
                   
                 
                 
                   
                     
                       ( 
                       
                         2 
                          
                         π 
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                    
                   
                     a 
                     
                       n 
                       1 
                     
                   
                 
               
                
               
                 
                   P 
                   
                     
                       n 
                       2 
                     
                     - 
                     
                       1 
                       2 
                     
                   
                   
                     
                       1 
                       2 
                     
                     - 
                     
                       n 
                       1 
                     
                   
                 
                  
                 
                   ( 
                   
                     cos 
                      
                     
                         
                     
                      
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
         [0161]    For the special case n 1 =n 2 =n 3 =0 it reads: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                   t 
                   1 
                 
                  
                 
                   
                     J 
                     0 
                   
                    
                   
                     ( 
                     at 
                     ) 
                   
                 
                  
                 
                   
                     J 
                     
                       n 
                       2 
                     
                   
                    
                   
                     ( 
                     bt 
                     ) 
                   
                 
                  
                 
                   
                     J 
                     
                       n 
                       2 
                     
                   
                    
                   
                     ( 
                     ct 
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                    
                   t 
                 
               
             
             = 
             
               
                 
                   
                     
                       ( 
                       bc 
                       ) 
                     
                     
                       - 
                       1 
                     
                   
                    
                   
                     
                       sin 
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                      
                     
                       ( 
                       A 
                       ) 
                     
                   
                 
                 
                   
                     ( 
                     
                       2 
                        
                       π 
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
                
               
                 
                   P 
                   
                     
                       n 
                       2 
                     
                     - 
                     
                       1 
                       2 
                     
                   
                   
                     1 
                     2 
                   
                 
                  
                 
                   ( 
                   
                     cos 
                      
                     
                         
                     
                      
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
         [0162]    For our case a=b=k 0 , c=|k| we get: 
         [0000]    
       
         
           
             
               
                 
                   
                     cos 
                      
                     
                         
                     
                      
                     A 
                   
                   = 
                   
                     
                       
                         b 
                         2 
                       
                       + 
                       
                         c 
                         2 
                       
                       - 
                       
                         a 
                         2 
                       
                     
                     
                       2 
                        
                       bc 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         k 
                         0 
                         2 
                       
                       + 
                       
                         
                            
                           k 
                            
                         
                         2 
                       
                       - 
                       
                         k 
                         0 
                         2 
                       
                     
                     
                       2 
                        
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                        
                       k 
                        
                     
                     
                       2 
                        
                       
                         k 
                         0 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   z 
                 
               
             
           
         
       
       
         
           
             
               sin 
                
               
                 ( 
                 A 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   - 
                   
                     
                       cos 
                       2 
                     
                      
                     A 
                   
                 
               
               = 
               
                 
                   
                     1 
                     - 
                     
                       z 
                       2 
                     
                   
                 
                 = 
                 
                   
                     
                       1 
                       - 
                       
                         
                           
                              
                             k 
                              
                           
                           2 
                         
                         
                           4 
                            
                           
                             k 
                             0 
                             2 
                           
                         
                       
                     
                   
                   . 
                   
                     
 
                   
                    
                   Therefore 
                 
               
             
           
         
       
       
         
           
             Δ 
             = 
             
               
                 
                   1 
                   2 
                 
                  
                 bc 
                  
                 
                     
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     A 
                     ) 
                   
                 
               
               = 
               
                 
                   1 
                   2 
                 
                  
                 
                   k 
                   0 
                 
                  
                 
                    
                   k 
                    
                 
                  
                 
                   
                     1 
                     - 
                     
                       
                         
                            
                           k 
                            
                         
                         2 
                       
                       
                         4 
                          
                         
                           k 
                           0 
                           2 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       sin 
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                      
                     
                       ( 
                       A 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           1 
                           - 
                           
                             z 
                             2 
                           
                         
                       
                       ) 
                     
                     
                       - 
                       
                         1 
                         2 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       [ 
                       
                         
                           ( 
                           
                             1 
                             + 
                             z 
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             z 
                           
                           ) 
                         
                       
                       ] 
                     
                     
                       - 
                       
                         1 
                         4 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     &gt; 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         t 
                          
                         
                             
                         
                          
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               t 
                             
                             ) 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               t 
                             
                             ) 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               
                                  
                                 k 
                                  
                               
                                
                               t 
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           t 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         
                           
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   π 
                                 
                                 ) 
                               
                               
                                 1 
                                 2 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     z 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     z 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                           
                             1 
                             4 
                           
                         
                       
                     
                      
                     
                       
                         P 
                         
                           
                             n 
                             2 
                           
                           - 
                           
                             1 
                             2 
                           
                         
                         
                           1 
                           2 
                         
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0163]    We now look at the definition of the Legendre Functions: 
         [0000]    
       
         
           
             
               
                 P 
                 λ 
                 μ 
               
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 
                   
                     1 
                     
                       Γ 
                        
                       
                         ( 
                         
                           1 
                           - 
                           μ 
                         
                         ) 
                       
                     
                   
                    
                   
                     [ 
                     
                       
                         1 
                         + 
                         z 
                       
                       
                         1 
                         - 
                         z 
                       
                     
                     ] 
                   
                 
                 
                   μ 
                   2 
                 
               
                
               
                 
                   F 
                   1 
                     
                     
                   2 
                     
                 
                  
                 
                   ( 
                   
                     
                       - 
                       λ 
                     
                     , 
                     
                       λ 
                       + 
                       1 
                     
                     , 
                     
                       1 
                       - 
                       μ 
                     
                     , 
                     
                       
                         1 
                         - 
                         z 
                       
                       2 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               Using 
                
               
                   
               
                
               
                 Γ 
                  
                 
                   ( 
                   
                     1 
                     2 
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   π 
                 
                  
                 
                     
                 
                  
                 we 
                  
                 
                     
                 
                  
                 get 
                  
                 
                     
                 
                  
                 for 
                  
                 
                     
                 
                  
                 μ 
               
               = 
               
                 
                   
                     1 
                     2 
                   
                    
                   
                       
                   
                    
                   and 
                    
                   
                       
                   
                    
                   λ 
                 
                 = 
                 
                   
                     
                       
                         n 
                         2 
                       
                       - 
                       
                         1 
                         2 
                       
                     
                     : 
                     
                       
 
                     
                      
                     
                       
                         P 
                         
                           
                             n 
                             2 
                           
                           - 
                           
                             1 
                             2 
                           
                         
                         
                           1 
                           2 
                         
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           1 
                           
                             π 
                           
                         
                          
                         
                           [ 
                           
                             
                               1 
                               + 
                               z 
                             
                             
                               1 
                               - 
                               z 
                             
                           
                           ] 
                         
                       
                       
                         1 
                         4 
                       
                     
                      
                     
                       
                         F 
                         1 
                           
                           
                         2 
                           
                       
                        
                       
                         ( 
                         
                           
                             
                               - 
                               
                                 n 
                                 2 
                               
                             
                             + 
                             
                               1 
                               2 
                             
                           
                           , 
                           
                             
                               n 
                               2 
                             
                             - 
                             
                               1 
                               2 
                             
                             + 
                             1 
                           
                           , 
                           
                             1 
                             2 
                           
                           , 
                           
                             
                               1 
                               - 
                               z 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                      
                     
                       t 
                        
                       
                           
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         t 
                       
                     
                   
                   = 
                     
                    
                   
                     1 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       
                         
                           
                             
                               ( 
                               
                                 2 
                                  
                                 π 
                               
                               ) 
                             
                             
                               1 
                               2 
                             
                           
                            
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   z 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   z 
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                         
                           1 
                           4 
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         
                           1 
                           
                             π 
                           
                         
                          
                         
                           [ 
                           
                             
                               1 
                               + 
                               z 
                             
                             
                               1 
                               - 
                               z 
                             
                           
                           ] 
                         
                       
                       
                         1 
                         4 
                       
                     
                      
                     
                       F 
                       1 
                         
                         
                       2 
                         
                     
                   
                 
               
             
             
               
                 
                     
                    
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               + 
                               
                                 1 
                                 2 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               
                                 n 
                                 2 
                               
                               - 
                               
                                 1 
                                 2 
                               
                               + 
                               1 
                             
                             , 
                             
                               1 
                               2 
                             
                             , 
                             
                               
                                 1 
                                 - 
                                 z 
                               
                               2 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       1 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         
                           2 
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             z 
                           
                         
                       
                     
                      
                     
                       F 
                       1 
                         
                         
                       2 
                         
                     
                   
                 
               
             
             
               
                 
                     
                    
                   
                     ( 
                     
                       
                         
                           - 
                           
                             n 
                             2 
                           
                         
                         + 
                         
                           1 
                           2 
                         
                       
                       , 
                       
                         
                           n 
                           2 
                         
                         + 
                         
                           1 
                           2 
                         
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             ( 
             
               
                 
                   F 
                   1 
                     
                     
                   2 
                     
                 
                  
                 
                   ( 
                   
                     
                       1 
                       2 
                     
                     , 
                     
                       
                         1 
                         2 
                       
                       ; 
                       
                         1 
                         2 
                       
                       ; 
                       z 
                     
                   
                   ) 
                 
               
               = 
               
                 1 
                 
                   
                     1 
                     - 
                     z 
                   
                 
               
             
             ) 
           
         
       
       
         
           
             
               
                 
                   F 
                   1 
                     
                     
                   2 
                     
                 
                  
                 
                   ( 
                   
                     
                       1 
                       2 
                     
                     , 
                     
                       1 
                       2 
                     
                     , 
                     
                       1 
                       2 
                     
                     , 
                     y 
                   
                   ) 
                 
               
               = 
               
                 1 
                 
                   
                     1 
                     - 
                     y 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   
                     
                         
                     
                     2 
                   
                    
                   
                     
                       F 
                       1 
                     
                      
                     
                       ( 
                       
                         
                           1 
                           2 
                         
                         , 
                         
                           1 
                           2 
                         
                         , 
                         
                           1 
                           2 
                         
                         , 
                         
                           
                             1 
                             - 
                             z 
                           
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     1 
                     
                       
                         1 
                         - 
                         
                           
                             1 
                             - 
                             z 
                           
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       2 
                     
                     
                       
                         2 
                         - 
                         
                           ( 
                           
                             1 
                             - 
                             z 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       2 
                     
                     
                       
                         1 
                         + 
                         z 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                     
                   
                     = 
                     
                       
                         
                           &gt; 
                           2 
                         
                          
                         
                           
                             F 
                             1 
                           
                            
                           
                             ( 
                             
                               
                                 1 
                                 2 
                               
                               , 
                               
                                 1 
                                 2 
                               
                               , 
                               
                                 1 
                                 2 
                               
                               , 
                               
                                 
                                   1 
                                   - 
                                   z 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       = 
                         
                        
                       
                         1 
                         
                           
                             1 
                             - 
                             
                               
                                 1 
                                 - 
                                 z 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       2 
                     
                     
                       
                         2 
                         - 
                         
                           ( 
                           
                             1 
                             - 
                             z 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       2 
                     
                     
                       
                         1 
                         + 
                         z 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                      
                     
                       t 
                        
                       
                           
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         t 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       1 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         
                           2 
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             z 
                           
                         
                       
                     
                      
                     
                       F 
                       1 
                         
                         
                       2 
                         
                     
                   
                 
               
             
             
               
                 
                     
                    
                   
                     ( 
                     
                       
                         1 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       1 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         
                           2 
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             z 
                           
                         
                       
                     
                      
                     
                       
                         2 
                       
                       
                         
                           1 
                           + 
                           z 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       1 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             z 
                           
                         
                       
                     
                      
                     
                       1 
                       
                         
                           1 
                           + 
                           z 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       π 
                        
                       
                         
                           [ 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 z 
                                 2 
                               
                             
                             ) 
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       π 
                        
                       
                         
                           1 
                           - 
                           
                             
                               
                                  
                                 k 
                                  
                               
                               2 
                             
                             
                               4 
                                
                               
                                 k 
                                 0 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0164]    For n 2 =0 the Gauss Hypergeometric Function is found in 9    
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       2 
                     
                      
                     
                       
                         
                           F 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               1 
                               2 
                             
                             , 
                             
                               
                                 1 
                                 2 
                               
                               : 
                               
                                 
                                   1 
                                   2 
                                 
                                 : 
                                 z 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         1 
                         
                           
                             1 
                             - 
                             z 
                           
                         
                       
                     
                     ) 
                   
                    
                   
                     
                       F 
                       1 
                         
                         
                       2 
                         
                     
                      
                     
                       ( 
                       
                         
                           1 
                           2 
                         
                         , 
                         
                           1 
                           2 
                         
                         , 
                         
                           1 
                           2 
                         
                         , 
                         y 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   1 
                   
                     
                       1 
                       - 
                       z 
                     
                   
                 
               
               ; 
             
             = 
             
               
                 &gt; 
                 
                   
                     F 
                     1 
                       
                       
                     2 
                       
                   
                    
                   
                     ( 
                     
                       
                         1 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   1 
                   
                     
                       1 
                       - 
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                     
                     
                       
                         2 
                         - 
                         
                           ( 
                           
                             1 
                             - 
                             z 
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           2 
                         
                         
                           
                             1 
                             + 
                             z 
                           
                         
                       
                        
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                          
                         
                           t 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 t 
                               
                               ) 
                             
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 t 
                               
                               ) 
                             
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 t 
                               
                               ) 
                             
                           
                            
                           
                              
                             t 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               k 
                                
                             
                              
                             
                               2 
                             
                              
                             π 
                              
                             
                               
                                 1 
                                 - 
                                 z 
                               
                             
                           
                         
                          
                         
                           
                             F 
                             1 
                               
                               
                             2 
                               
                           
                            
                           
                             ( 
                             
                               
                                 1 
                                 2 
                               
                               , 
                               
                                 1 
                                 2 
                               
                               , 
                               
                                 1 
                                 2 
                               
                               , 
                               
                                 
                                   1 
                                   - 
                                   z 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             1 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 k 
                                  
                               
                                
                               
                                 2 
                               
                                
                               π 
                                
                               
                                 
                                   1 
                                   - 
                                   z 
                                 
                               
                             
                           
                            
                           
                             
                               2 
                             
                             
                               
                                 1 
                                 + 
                                 z 
                               
                             
                           
                         
                         = 
                         
                           
                             
                               1 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 π 
                                  
                                 
                                   
                                     1 
                                     - 
                                     z 
                                   
                                 
                               
                             
                              
                             
                               1 
                               
                                 
                                   
                                     1 
                                     + 
                                     z 
                                   
                                   ) 
                                 
                               
                             
                           
                           = 
                           
                             
                               1 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 π 
                                  
                                 
                                   
                                     [ 
                                     
                                       ( 
                                       
                                         1 
                                         - 
                                         
                                           z 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ] 
                                   
                                 
                               
                             
                             = 
                             
                               1 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                   
                                     1 
                                     - 
                                     
                                       
                                         
                                            
                                           k 
                                            
                                         
                                         2 
                                       
                                       
                                         4 
                                          
                                         
                                           k 
                                           0 
                                           2 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]      9  http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/07/07/01/
 
we used
 
         [0000]    
       
         
           
             z 
             = 
             
               
                 
                    
                   k 
                    
                 
                 
                   2 
                    
                   
                     k 
                     0 
                   
                 
               
               . 
             
           
         
       
     
         [0000]    n 2 =1 
         [0165]    The above Gauss Hypergeometric Function is 
         [0000]    
       
         
           
             
                 
             
              
             
               ( 
               
                 
                   
                     F 
                     1 
                       
                       
                     2 
                       
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                       , 
                       
                         
                           3 
                           2 
                         
                         ; 
                         
                           1 
                           2 
                         
                         ; 
                         z 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     1 
                     - 
                     
                       2 
                        
                       z 
                     
                   
                   
                     
                       1 
                       - 
                       z 
                     
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               
                 
                   
                       
                     
                         
                     
                   
                   2 
                 
                  
                 
                   
                     F 
                     1 
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                       , 
                       
                         3 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       y 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   1 
                   - 
                   
                     2 
                      
                     y 
                   
                 
                 
                   
                     1 
                     - 
                     y 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 =&gt; 
               
                
               
                 
                     
                   
                       
                   
                 
                 2 
               
                
               
                 
                   F 
                   1 
                 
                  
                 
                   ( 
                   
                     
                       - 
                       
                         1 
                         2 
                       
                     
                     , 
                     
                       3 
                       2 
                     
                     , 
                     
                       1 
                       2 
                     
                     , 
                     
                       
                         1 
                         - 
                         z 
                       
                       2 
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 z 
                 
                   
                     1 
                     - 
                     
                       
                         1 
                         - 
                         z 
                       
                       2 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       2 
                     
                      
                     z 
                   
                   
                     
                       2 
                       - 
                       
                         ( 
                         
                           1 
                           - 
                           z 
                         
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                     
                      
                     z 
                   
                   
                     
                       1 
                       + 
                       z 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 t 
                  
                 
                     
                 
                  
                 
                   
                     J 
                     0 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       t 
                     
                     ) 
                   
                 
                  
                 
                   
                     J 
                     1 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       t 
                     
                     ) 
                   
                 
                  
                 
                   
                     J 
                     1 
                   
                    
                   
                     ( 
                     
                       
                          
                         k 
                          
                       
                        
                       t 
                     
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                    
                   t 
                 
               
             
             = 
             
               
                 
                   1 
                   
                     
                       k 
                       0 
                     
                      
                     
                        
                       k 
                        
                     
                      
                     
                       2 
                     
                      
                     π 
                      
                     
                       
                         1 
                         - 
                         z 
                       
                     
                   
                 
                  
                 
                   
                     F 
                     1 
                       
                       
                     2 
                       
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                       , 
                       
                         3 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     
                       1 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         
                           2 
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             z 
                           
                         
                       
                     
                      
                     
                       
                         
                           2 
                         
                          
                         z 
                       
                       
                         
                           1 
                           + 
                           z 
                         
                       
                     
                   
                   == 
                   
                     z 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       π 
                        
                       
                         
                           1 
                           - 
                           
                             z 
                             2 
                           
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       
                          
                         k 
                          
                       
                       
                         2 
                          
                         
                           k 
                           0 
                         
                       
                     
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       π 
                        
                       
                         
                           1 
                           - 
                           
                             
                               
                                  
                                 k 
                                  
                               
                               2 
                             
                             
                               4 
                                
                               
                                 k 
                                 0 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     1 
                     
                       2 
                        
                       
                         k 
                         0 
                         2 
                       
                        
                       π 
                        
                       
                         
                           1 
                           - 
                           
                             
                               
                                  
                                 k 
                                  
                               
                               2 
                             
                             
                               4 
                                
                               
                                 k 
                                 0 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    we used 
         [0000]    
       
         
           
             z 
             = 
             
               
                 
                    
                   k 
                    
                 
                 
                   2 
                    
                   
                     k 
                     0 
                   
                 
               
               . 
             
           
         
       
     
         [0000]    n 2 =2 
         [0166]    The above Gauss Hypergeometric Function is 
         [0000]    
       
         
           
             
                 
             
              
             
               ( 
               
                 
                   
                     F 
                     1 
                       
                       
                     2 
                       
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           3 
                           2 
                         
                       
                       , 
                       
                         
                           5 
                           2 
                         
                         ; 
                         
                           1 
                           2 
                         
                         ; 
                         z 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       8 
                        
                       
                         ( 
                         
                           z 
                           - 
                           3 
                         
                         ) 
                       
                        
                       z 
                     
                     + 
                     1 
                   
                   
                     
                       1 
                       - 
                       z 
                     
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               
                 
                   
                       
                     
                         
                     
                   
                   2 
                 
                  
                 
                   
                     F 
                     1 
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           3 
                           2 
                         
                       
                       , 
                       
                         5 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       y 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     8 
                      
                     
                       ( 
                       
                         y 
                         - 
                         1 
                       
                       ) 
                     
                      
                     y 
                   
                   + 
                   1 
                 
                 
                   
                     1 
                     - 
                     y 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 =&gt; 
               
                
               
                   
               
                
               
                 
                   F 
                   1 
                     
                     
                   2 
                     
                 
                  
                 
                   ( 
                   
                     
                       - 
                       
                         3 
                         2 
                       
                     
                     , 
                     
                       5 
                       2 
                     
                     , 
                     
                       1 
                       2 
                     
                     , 
                     
                       
                         1 
                         - 
                         z 
                       
                       2 
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   
                     8 
                      
                     
                       ( 
                       
                         
                           
                             1 
                             - 
                             z 
                           
                           2 
                         
                         - 
                         1 
                       
                       ) 
                     
                      
                     
                       
                         1 
                         - 
                         z 
                       
                       2 
                     
                   
                   + 
                   1 
                 
                 
                   
                     1 
                     - 
                     
                       
                         1 
                         - 
                         z 
                       
                       2 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       
                         - 
                         2 
                       
                        
                       
                         ( 
                         
                           z 
                           + 
                           1 
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           z 
                         
                         ) 
                       
                     
                     + 
                     1 
                   
                   
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             1 
                             - 
                             z 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       
                         2 
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                             2 
                              
                             
                               ( 
                               
                                 z 
                                 + 
                                 1 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 z 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     
                       
                         1 
                         + 
                         z 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           2 
                         
                          
                         
                           [ 
                           
                             1 
                             + 
                             
                               2 
                                
                               
                                 ( 
                                 
                                   
                                     z 
                                     2 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       
                         
                           1 
                           + 
                           z 
                         
                       
                     
                     = 
                     
                       
                         
                           2 
                         
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 z 
                                 2 
                               
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                       
                         
                           1 
                           + 
                           z 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 t 
                  
                 
                     
                 
                  
                 
                   
                     J 
                     0 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       t 
                     
                     ) 
                   
                 
                  
                 
                   
                     J 
                     2 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       t 
                     
                     ) 
                   
                 
                  
                 
                   
                     J 
                     2 
                   
                    
                   
                     ( 
                     
                       
                          
                         k 
                          
                       
                        
                       t 
                     
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                    
                   t 
                 
               
             
             = 
             
               
                 
                   1 
                   
                     
                       k 
                       0 
                     
                      
                     
                        
                       k 
                        
                     
                      
                     
                       2 
                     
                      
                     π 
                      
                     
                       
                         1 
                         - 
                         z 
                       
                     
                   
                 
                  
                 
                   
                     F 
                     1 
                       
                       
                     2 
                       
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           3 
                           2 
                         
                       
                       , 
                       
                         5 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     1 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       
                         2 
                       
                        
                       π 
                        
                       
                         
                           1 
                           - 
                           z 
                         
                       
                     
                   
                    
                   
                     
                       
                         2 
                       
                        
                       
                         ( 
                         
                           
                             2 
                              
                             
                               z 
                               2 
                             
                           
                           - 
                           1 
                         
                         ) 
                       
                     
                     
                       
                         1 
                         + 
                         z 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       
                         2 
                          
                         
                           z 
                           2 
                         
                       
                       - 
                       1 
                     
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       π 
                        
                       
                         
                           1 
                           - 
                           
                             z 
                             2 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           2 
                            
                           
                             z 
                             2 
                           
                         
                         - 
                         1 
                       
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             
                               z 
                               2 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                                
                               k 
                                
                             
                             2 
                           
                           
                             2 
                              
                             
                               k 
                               0 
                               2 
                             
                           
                         
                         - 
                         1 
                       
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             
                               
                                 
                                    
                                   k 
                                    
                                 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   k 
                                   0 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    we used 
         [0000]    
       
         
           
             z 
             = 
             
               
                 
                    
                   k 
                    
                 
                 
                   2 
                    
                   
                     k 
                     0 
                   
                 
               
               . 
             
           
         
       
     
         [0000]    n 2 =3 
         [0167]    The above Gauss Hypergeometric Function is 
         [0000]    
       
         
           
             
                 
             
              
             
               ( 
               
                 
                   
                     F 
                     1 
                       
                       
                     2 
                       
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           5 
                           2 
                         
                       
                       , 
                       
                         
                           7 
                           2 
                         
                         ; 
                         
                           1 
                           2 
                         
                         ; 
                         z 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     1 
                     - 
                     
                       2 
                        
                       
                         
                           ( 
                           
                             3 
                             - 
                             
                               4 
                                
                               z 
                             
                           
                           ) 
                         
                         2 
                       
                        
                       z 
                     
                   
                   
                     
                       1 
                       - 
                       z 
                     
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               
                 F 
                 1 
                   
                   
                 2 
                   
               
                
               
                 ( 
                 
                   
                     - 
                     
                       5 
                       2 
                     
                   
                   , 
                   
                     7 
                     2 
                   
                   , 
                   
                     1 
                     2 
                   
                   , 
                   y 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     1 
                     - 
                     
                       2 
                        
                       
                         
                           ( 
                           
                             3 
                             - 
                             
                               4 
                                
                               y 
                             
                           
                           ) 
                         
                         2 
                       
                        
                       y 
                     
                   
                   
                     
                       1 
                       - 
                       y 
                     
                   
                 
                  
                 
                   =&gt; 
                 
                  
                 
                   
                     F 
                     1 
                       
                       
                     
                       
                           
                       
                        
                       2 
                     
                       
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           5 
                           2 
                         
                       
                       , 
                       
                         7 
                         2 
                       
                       , 
                       
                         1 
                         2 
                       
                       , 
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     1 
                     - 
                     
                       2 
                        
                       
                         
                           ( 
                           
                             3 
                             - 
                             
                               4 
                                
                               
                                 
                                   1 
                                   - 
                                   z 
                                 
                                 2 
                               
                             
                           
                           ) 
                         
                         2 
                       
                        
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                   
                   
                     
                       1 
                       - 
                       
                         
                           1 
                           - 
                           z 
                         
                         2 
                       
                     
                   
                 
                 == 
                 
                   
                     
                       2 
                     
                      
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   2 
                                    
                                   z 
                                 
                               
                               ) 
                             
                             2 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               z 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   
                     
                       1 
                       + 
                       z 
                     
                   
                 
               
             
           
         
       
     
         [0168]    We get: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                      
                     
                       t 
                        
                       
                           
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                         
                           J 
                           3 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                         
                           J 
                           3 
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             t 
                           
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         t 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       1 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         
                           2 
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             z 
                           
                         
                       
                     
                      
                     
                       
                         F 
                         1 
                           
                           
                         
                           
                               
                           
                            
                           2 
                         
                           
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               5 
                               2 
                             
                           
                           , 
                           
                             7 
                             2 
                           
                           , 
                           
                             1 
                             2 
                           
                           , 
                           
                             
                               1 
                               - 
                               z 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       1 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           k 
                            
                         
                          
                         
                           2 
                         
                          
                         π 
                          
                         
                           
                             1 
                             - 
                             z 
                           
                         
                       
                     
                      
                     
                       
                         
                           2 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       2 
                                        
                                       z 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   z 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       
                         
                           1 
                           + 
                           z 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       1 
                       - 
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 2 
                                  
                                 z 
                               
                             
                             ) 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             z 
                           
                           ) 
                         
                       
                     
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       π 
                        
                       
                         
                           1 
                           - 
                           
                             z 
                             2 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       1 
                       - 
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                    
                                   k 
                                    
                                 
                                 
                                   k 
                                   0 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                  
                                 k 
                                  
                               
                               
                                 2 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         k 
                          
                       
                        
                       π 
                        
                       
                         
                           1 
                           - 
                           
                             
                               
                                  
                                 k 
                                  
                               
                               2 
                             
                             
                               4 
                                
                               
                                 k 
                                 0 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    we used 
         [0000]    
       
         
           
             z 
             = 
             
               
                 
                    
                   k 
                    
                 
                 
                   2 
                    
                   
                       
                   
                    
                   
                     k 
                     0 
                   
                 
               
               . 
             
           
         
       
     
       Annex A II 
     Integration to obtain H BF (k) 
       [0169]        h   BF ( z−r )= N∫   0   ξ   dφ   r ∫ 0   ξ   dφ   t ×exp{− ik   0   û (φ r )·( z−r )}exp{ ik   0   û (φ t )·( z−r )}
 
         [0170]    Bessel Functions 10  and Jacobi Anger Expansion 11 : 
         [0000]    
       
         
           
             
               
                 
                   
                     〈 
                     
                       
                         r 
                         - 
                         z 
                       
                       | 
                       
                         k 
                         0 
                       
                     
                     〉 
                   
                   = 
                     
                    
                   
                     exp 
                      
                     
                       [ 
                       
                         
                           k 
                           0 
                         
                         · 
                         
                           ( 
                           
                             r 
                             - 
                             z 
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     exp 
                      
                     
                       { 
                       
                          
                          
                         
                             
                         
                          
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             α 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           - 
                           ∞ 
                         
                       
                       ∞ 
                     
                      
                     
                       
                          
                         n 
                       
                        
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                              
                              
                             
                                 
                             
                              
                             n 
                              
                             
                                 
                             
                              
                             α 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           - 
                           ∞ 
                         
                       
                       ∞ 
                     
                      
                     
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         exp 
                          
                         
                           [ 
                           
                              
                              
                             
                                 
                             
                              
                             n 
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 α 
                                 + 
                                 
                                   π 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         J 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           , 
                           
                             n 
                             ≠ 
                             0 
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           exp 
                            
                           
                             [ 
                             
                                
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   α 
                                   + 
                                   
                                     π 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             α 
             = 
             
               
                 φ 
                 
                   r 
                   , 
                   t 
                 
               
               - 
               
                 
                   φ 
                   ′ 
                 
                 . 
               
             
           
         
       
     
         [0000]    Integration over φ r : 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 ξ 
               
                
               
                   
               
                
               
                 
                    
                   
                     φ 
                     r 
                   
                 
                  
                 exp 
                  
                 
                   { 
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     
                       k 
                       0 
                     
                      
                     
                       
                         
                           u 
                           ^ 
                         
                          
                         
                           ( 
                           
                             φ 
                             r 
                           
                           ) 
                         
                       
                       · 
                       
                         ( 
                         
                           z 
                           - 
                           r 
                         
                         ) 
                       
                     
                   
                   } 
                 
               
             
             = 
             
               
                 
                   ξ 
                    
                   
                       
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     ∫ 
                     0 
                     ξ 
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         r 
                       
                     
                      
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             , 
                             
                               n 
                               ≠ 
                               0 
                             
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           exp 
                            
                           
                             [ 
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     + 
                                     
                                       π 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               = 
               
                 
                   
                     ξ 
                      
                     
                         
                     
                      
                     
                       
                         J 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     
                       ∫ 
                       0 
                       ξ 
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           φ 
                           r 
                         
                       
                        
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 n 
                                 ≠ 
                                 0 
                               
                             
                           
                           ∞ 
                         
                          
                         
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             exp 
                              
                             
                               [ 
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       
                                         φ 
                                         r 
                                       
                                       - 
                                       
                                         φ 
                                         ′ 
                                       
                                       + 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ξ 
                        
                       
                           
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             , 
                             
                               n 
                               ≠ 
                               0 
                             
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             ∫ 
                             0 
                             ξ 
                           
                            
                           
                               
                           
                            
                           
                             
                                
                               
                                 φ 
                                 r 
                               
                             
                              
                             
                               exp 
                                
                               
                                 [ 
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           r 
                                         
                                         - 
                                         
                                           φ 
                                           ′ 
                                         
                                         + 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ξ 
                          
                         
                             
                         
                          
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 n 
                                 ≠ 
                                 0 
                               
                             
                           
                           ∞ 
                         
                          
                         
                           
                             exp 
                              
                             
                               [ 
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       
                                         - 
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               ∫ 
                               0 
                               ξ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   φ 
                                   r 
                                 
                               
                                
                               
                                 exp 
                                  
                                 
                                   [ 
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       r 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ξ 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             
                               ∑ 
                               
                                 
                                   n 
                                   = 
                                   
                                     - 
                                     ∞ 
                                   
                                 
                                 , 
                                 
                                   n 
                                   ≠ 
                                   0 
                                 
                               
                             
                             ∞ 
                           
                            
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       
                                         φ 
                                         ′ 
                                       
                                       - 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                              
                             
                               
                                 J 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                                 - 
                                 1 
                               
                               
                                 
                                   - 
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   
                                      
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           ξ 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             
                               ∑ 
                               
                                 
                                   n 
                                   = 
                                   
                                     - 
                                     ∞ 
                                   
                                 
                                 , 
                                 
                                   n 
                                   ≠ 
                                   0 
                                 
                               
                             
                             ∞ 
                           
                            
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                     
                                        
                                        
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     
                                       π 
                                       2 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             n 
                                              
                                             
                                                 
                                             
                                              
                                             ξ 
                                           
                                         
                                         - 
                                         1 
                                       
                                       ] 
                                     
                                   
                                 
                               
                               
                                 - 
                                 n 
                               
                             
                              
                             
                               
                                 J 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 
                                   φ 
                                   ′ 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]      10  http://en.wikipedia.org/wiki/Bessel_function 11  http://en.wikipedia.org/wiki/Jacobi-Anger_expansion 
         [0171]    Integration over φ t : 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 ξ 
               
                
               
                   
               
                
               
                 
                    
                   
                     φ 
                     t 
                   
                 
                  
                 exp 
                  
                 
                   { 
                   
                      
                      
                     
                         
                     
                      
                     
                       k 
                       0 
                     
                      
                     
                       
                         
                           u 
                           ^ 
                         
                          
                         
                           ( 
                           
                             φ 
                             r 
                           
                           ) 
                         
                       
                       · 
                       
                         ( 
                         
                           z 
                           - 
                           r 
                         
                         ) 
                       
                     
                   
                   } 
                 
               
             
             = 
             
               
                 
                   ξ 
                    
                   
                       
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     ∫ 
                     0 
                     ξ 
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         t 
                       
                     
                      
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             , 
                             
                               n 
                               ≠ 
                               0 
                             
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           exp 
                            
                           
                             [ 
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     + 
                                     
                                       π 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               = 
               
                 
                   
                     ξ 
                      
                     
                         
                     
                      
                     
                       
                         J 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     
                       ∫ 
                       0 
                       ξ 
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           φ 
                           t 
                         
                       
                        
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 n 
                                 ≠ 
                                 0 
                               
                             
                           
                           ∞ 
                         
                          
                         
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             exp 
                              
                             
                               [ 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       
                                         φ 
                                         t 
                                       
                                       - 
                                       
                                         φ 
                                         ′ 
                                       
                                       + 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ξ 
                        
                       
                           
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             , 
                             
                               n 
                               ≠ 
                               0 
                             
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             ∫ 
                             0 
                             ξ 
                           
                            
                           
                               
                           
                            
                           
                             
                                
                               
                                 φ 
                                 t 
                               
                             
                              
                             
                               exp 
                                
                               
                                 [ 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           t 
                                         
                                         - 
                                         
                                           φ 
                                           ′ 
                                         
                                         + 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ξ 
                          
                         
                             
                         
                          
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 n 
                                 ≠ 
                                 0 
                               
                             
                           
                           ∞ 
                         
                          
                         
                           
                             exp 
                              
                             
                               [ 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       
                                         - 
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               ∫ 
                               0 
                               ξ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   φ 
                                   t 
                                 
                               
                                
                               
                                 exp 
                                  
                                 
                                   [ 
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       t 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ξ 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             
                               ∑ 
                               
                                 
                                   n 
                                   = 
                                   
                                     - 
                                     ∞ 
                                   
                                 
                                 , 
                                 
                                   n 
                                   ≠ 
                                   0 
                                 
                               
                             
                             ∞ 
                           
                            
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       
                                         - 
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                              
                             
                               
                                 J 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                                 - 
                                 1 
                               
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           
                             ξ 
                              
                             
                                 
                             
                              
                             
                               
                                 J 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     = 
                                     
                                       - 
                                       ∞ 
                                     
                                   
                                   , 
                                   
                                     n 
                                     ≠ 
                                     0 
                                   
                                 
                               
                               ∞ 
                             
                              
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                      
                                     
                                       ( 
                                       
                                         
                                           - 
                                           
                                             φ 
                                             ′ 
                                           
                                         
                                         + 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                                
                               
                                 
                                   J 
                                   n 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                   
                                   - 
                                   1 
                                 
                                 
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                    
                                   
                                       
                                   
                                 
                               
                                
                               
                                  
                                  
                               
                             
                           
                         
                         = 
                         
                           
                             
                               ξ 
                                
                               
                                   
                               
                                
                               
                                 
                                   J 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   ∑ 
                                   
                                     
                                       n 
                                       = 
                                       
                                         - 
                                         ∞ 
                                       
                                     
                                     , 
                                     
                                       n 
                                       ≠ 
                                       0 
                                     
                                   
                                 
                                 ∞ 
                               
                                
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                        
                                       
                                         ( 
                                         
                                           
                                             - 
                                             
                                               φ 
                                               ′ 
                                             
                                           
                                           + 
                                           
                                             π 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                  
                                 
                                   
                                     J 
                                     n 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         0 
                                       
                                        
                                       
                                          
                                         
                                           z 
                                           - 
                                           r 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                     - 
                                     1 
                                   
                                   
                                     
                                       - 
                                       
                                           
                                       
                                        
                                       n 
                                     
                                      
                                     
                                         
                                     
                                   
                                 
                               
                             
                           
                           = 
                           
                             
                               ξ 
                                
                               
                                   
                               
                                
                               
                                 
                                   J 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     = 
                                     
                                       - 
                                       ∞ 
                                     
                                   
                                   , 
                                   
                                     n 
                                     ≠ 
                                     0 
                                   
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   
                                     J 
                                     n 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         0 
                                       
                                        
                                       
                                          
                                         
                                           z 
                                           - 
                                           r 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                      
                                     
                                       
                                          
                                          
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           π 
                                           2 
                                         
                                          
                                         
                                           [ 
                                           
                                             
                                                
                                               
                                                  
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 n 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 ξ 
                                               
                                             
                                             - 
                                             1 
                                           
                                           ] 
                                         
                                       
                                     
                                   
                                   
                                     - 
                                     n 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       ′ 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0172]    Indeed the above integration over φ t  is the complex conjugate of the integration over φ r . 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 ⇒ 
                 
                   
 
                 
                  
                 
                   
                     h 
                     BF 
                   
                    
                   
                     ( 
                     
                       z 
                       - 
                       r 
                     
                     ) 
                   
                 
               
               = 
               
                 N 
                  
                 
                   { 
                   
                     
                       ξ 
                        
                       
                           
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             
                               n 
                               2 
                             
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           , 
                           
                             
                               n 
                               2 
                             
                             ≠ 
                             0 
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                           
                             n 
                             2 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                   
                   } 
                 
                  
                 
                   { 
                   
                     
                       ξ 
                        
                       
                           
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             
                               n 
                               1 
                             
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           , 
                           
                             
                               n 
                               1 
                             
                             ≠ 
                             0 
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       1 
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                           
                             n 
                             1 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                   
                   } 
                 
               
             
           
         
       
     
         [0173]    Fourier transforming: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             H 
                             BF 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           ∫ 
                           
                             
                               
                                  
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 h 
                                 BF 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   · 
                                   
                                     [ 
                                     
                                       z 
                                       - 
                                       r 
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           ∫ 
                           
                             
                               
                                  
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   · 
                                   
                                     [ 
                                     
                                       z 
                                       - 
                                       r 
                                     
                                     ] 
                                   
                                 
                               
                             
                             × 
                             × 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           N 
                            
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       ξ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           J 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         
                                           
                                             n 
                                             2 
                                           
                                           = 
                                           
                                             - 
                                             ∞ 
                                           
                                         
                                         , 
                                         
                                           
                                             n 
                                             2 
                                           
                                           ≠ 
                                           0 
                                         
                                       
                                       ∞ 
                                     
                                      
                                     
                                       
                                         
                                           
                                              
                                             
                                               
                                                 - 
                                                  
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   1 
                                                   + 
                                                   
                                                     n 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 π 
                                                 2 
                                               
                                             
                                           
                                            
                                           
                                             [ 
                                             
                                               1 
                                               - 
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                      
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     n 
                                                     2 
                                                   
                                                    
                                                   ξ 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         
                                           n 
                                           2 
                                         
                                       
                                        
                                       
                                         
                                           J 
                                           
                                             n 
                                             2 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         
                                            
                                            
                                           
                                               
                                           
                                            
                                           
                                             n 
                                             2 
                                           
                                            
                                           
                                             φ 
                                             ′ 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     ξ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         J 
                                         0 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                              
                                             
                                               z 
                                               - 
                                               r 
                                             
                                              
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       
                                         
                                           n 
                                           1 
                                         
                                         = 
                                         
                                           - 
                                           ∞ 
                                         
                                       
                                       , 
                                       
                                         
                                           n 
                                           1 
                                         
                                         ≠ 
                                         0 
                                       
                                     
                                     ∞ 
                                   
                                    
                                   
                                     
                                       
                                         
                                            
                                           
                                              
                                              
                                             
                                                 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   n 
                                                   1 
                                                 
                                               
                                               ) 
                                             
                                              
                                             
                                               π 
                                               2 
                                             
                                           
                                         
                                          
                                         
                                           [ 
                                           
                                             1 
                                             - 
                                             
                                                
                                               
                                                  
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   n 
                                                   1 
                                                 
                                                  
                                                 ξ 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       
                                         n 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         J 
                                         
                                           n 
                                           1 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                              
                                             
                                               z 
                                               - 
                                               r 
                                             
                                              
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           n 
                                           1 
                                         
                                          
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             H 
                             BF 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           ∫ 
                           
                             
                               
                                  
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 h 
                                 BF 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   · 
                                   
                                     [ 
                                     
                                       z 
                                       - 
                                       r 
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           ∫ 
                           
                             
                               
                                  
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   · 
                                   
                                     [ 
                                     
                                       z 
                                       - 
                                       r 
                                     
                                     ] 
                                   
                                 
                               
                             
                             × 
                             × 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           N 
                            
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ξ 
                                            
                                           
                                               
                                           
                                         
                                         2 
                                       
                                        
                                       
                                         
                                           J 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           J 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       ξ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           J 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         
                                           
                                             n 
                                             2 
                                           
                                           = 
                                           
                                             - 
                                             ∞ 
                                           
                                         
                                         , 
                                         
                                           
                                             n 
                                             2 
                                           
                                           ≠ 
                                           0 
                                         
                                       
                                       ∞ 
                                     
                                      
                                     
                                       
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                              
                                             
                                               π 
                                               2 
                                             
                                           
                                         
                                          
                                         
                                           [ 
                                           
                                             1 
                                             - 
                                             
                                                
                                               
                                                 
                                                   - 
                                                    
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   n 
                                                   2 
                                                 
                                                  
                                                 ξ 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       
                                         n 
                                         2 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           J 
                                           
                                             n 
                                             2 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         
                                            
                                            
                                           
                                               
                                           
                                            
                                           
                                             n 
                                             2 
                                           
                                            
                                           
                                             φ 
                                             ′ 
                                           
                                         
                                       
                                     
                                     + 
                                     
                                       ξ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           J 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           
                                             
                                               n 
                                               1 
                                             
                                             = 
                                             
                                               - 
                                               ∞ 
                                             
                                           
                                           , 
                                           
                                             
                                               n 
                                               1 
                                             
                                             ≠ 
                                             0 
                                           
                                         
                                         ∞ 
                                       
                                        
                                       
                                         
                                           
                                             
                                                
                                               
                                                  
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     + 
                                                     
                                                       n 
                                                       1 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   π 
                                                   2 
                                                 
                                               
                                             
                                              
                                             
                                               [ 
                                               
                                                 1 
                                                 - 
                                                 
                                                    
                                                   
                                                      
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       n 
                                                       1 
                                                     
                                                      
                                                     ξ 
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                           
                                             n 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             J 
                                             
                                               n 
                                               1 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 k 
                                                 0 
                                               
                                                
                                               
                                                  
                                                 
                                                   z 
                                                   - 
                                                   r 
                                                 
                                                  
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               n 
                                               1 
                                             
                                              
                                             
                                               φ 
                                               ′ 
                                             
                                           
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           ∑ 
                                           
                                             
                                               
                                                 n 
                                                 2 
                                               
                                               = 
                                               
                                                 - 
                                                 ∞ 
                                               
                                             
                                             , 
                                             
                                               
                                                 n 
                                                 2 
                                               
                                               ≠ 
                                               0 
                                             
                                           
                                           ∞ 
                                         
                                          
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                      
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       1 
                                                       + 
                                                       
                                                         n 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   
                                                     π 
                                                     2 
                                                   
                                                 
                                               
                                                
                                               
                                                 [ 
                                                 
                                                   1 
                                                   - 
                                                   
                                                      
                                                     
                                                       
                                                         - 
                                                          
                                                       
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         n 
                                                         2 
                                                       
                                                        
                                                       ξ 
                                                     
                                                   
                                                 
                                                 ] 
                                               
                                             
                                             
                                               n 
                                               2 
                                             
                                           
                                            
                                           
                                             
                                               J 
                                               
                                                 n 
                                                 2 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   0 
                                                 
                                                  
                                                 
                                                    
                                                   
                                                     z 
                                                     - 
                                                     r 
                                                   
                                                    
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                              
                                             
                                                
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 n 
                                                 2 
                                               
                                                
                                               
                                                 φ 
                                                 ′ 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     × 
                                   
                                 
                               
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         ∑ 
                                         
                                           
                                             
                                               n 
                                               1 
                                             
                                             = 
                                             
                                               - 
                                               ∞ 
                                             
                                           
                                           , 
                                           
                                             
                                               n 
                                               1 
                                             
                                             ≠ 
                                             0 
                                           
                                         
                                         ∞ 
                                       
                                        
                                       
                                         
                                           
                                             
                                                
                                               
                                                  
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     + 
                                                     
                                                       n 
                                                       1 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   π 
                                                   2 
                                                 
                                               
                                             
                                              
                                             
                                               [ 
                                               
                                                 1 
                                                 - 
                                                 
                                                    
                                                   
                                                      
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       n 
                                                       1 
                                                     
                                                      
                                                     ξ 
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                           
                                             n 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             J 
                                             
                                               n 
                                               1 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 k 
                                                 0 
                                               
                                                
                                               
                                                  
                                                 
                                                   z 
                                                   - 
                                                   r 
                                                 
                                                  
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               n 
                                               1 
                                             
                                              
                                             
                                               φ 
                                               ′ 
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≡ 
                           
                          
                         
                           ∫ 
                           
                             
                               
                                  
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   · 
                                   
                                     [ 
                                     
                                       z 
                                       - 
                                       r 
                                     
                                     ] 
                                   
                                 
                               
                             
                              
                             
                               { 
                               
                                 
                                   L 
                                   0 
                                 
                                 + 
                                 
                                   L 
                                   1 
                                 
                                 + 
                                 
                                   L 
                                   2 
                                 
                                 + 
                                 
                                   L 
                                   3 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0174]    The above expression is marked with intermediate terms for convenience. 
         [0175]    Inspect in L 1  the summation in two steps: 
         [0000]    
       
         
           
             
               
                 ∑ 
                 
                   
                     
                       n 
                       2 
                     
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   , 
                   
                     
                       n 
                       2 
                     
                     ≠ 
                     0 
                   
                 
                 ∞ 
               
                
               
                 
                   
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               n 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           π 
                           2 
                         
                       
                     
                      
                     
                       [ 
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             ξ 
                           
                         
                       
                       ] 
                     
                   
                   
                     n 
                     2 
                   
                 
                  
                 
                   
                     J 
                     
                       n 
                       2 
                     
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                     ) 
                   
                 
                  
                 
                    
                   
                      
                      
                     
                         
                     
                      
                     
                       n 
                       2 
                     
                      
                     
                       φ 
                       ′ 
                     
                   
                 
               
             
             = 
             
               
                 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         
                           
                             n 
                             2 
                           
                           = 
                           
                             - 
                             ∞ 
                           
                         
                         , 
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
                 == 
                 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                               - 
                               1 
                             
                             ] 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         
                           J 
                           
                             - 
                             
                               n 
                               2 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               = 
               
                 
                   
                     ∑ 
                     
                       
                         n 
                         2 
                       
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                       
                         n 
                         2 
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         
                           n 
                           2 
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                   
                 
                 + 
                 
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               
                                 n 
                                 2 
                               
                             
                             ) 
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                           
                           - 
                           1 
                         
                         ] 
                       
                     
                     
                       n 
                       2 
                     
                   
                    
                   
                     
                       J 
                       
                         - 
                         
                           n 
                           2 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         n 
                         2 
                       
                        
                       
                         φ 
                         ′ 
                       
                     
                   
                 
               
             
           
         
       
     
         [0176]    Using: J −n     2   (k 0 |z−r|)=(−1) n     2   J n     2   (k 0 |z−r|) we get: 
         [0000]    
       
         
           
             = 
             
               
                 
                   ∑ 
                   
                     
                       n 
                       2 
                     
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 n 
                                 2 
                               
                             
                             ) 
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                     
                       n 
                       2 
                     
                   
                    
                   
                     
                       J 
                       
                         n 
                         2 
                       
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       
                         n 
                         2 
                       
                        
                       
                         φ 
                         ′ 
                       
                     
                   
                 
               
               + 
               
                 
                   
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               n 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           π 
                           2 
                         
                       
                     
                      
                     
                       [ 
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             ξ 
                           
                         
                         - 
                         1 
                       
                       ] 
                     
                   
                   
                     n 
                     2 
                   
                 
                  
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     n 
                     2 
                   
                 
                  
                 
                   
                     J 
                     
                       n 
                       2 
                     
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                     ) 
                   
                 
                  
                 
                    
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     
                       n 
                       2 
                     
                      
                     
                       φ 
                       ′ 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               L 
               1 
             
             = 
             
               N 
                
               
                   
               
                
               ξ 
                
               
                   
               
                
               
                 
                   J 
                   0 
                 
                  
                 
                   ( 
                   
                     
                       k 
                       0 
                     
                      
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                   
                   ) 
                 
               
                
               
                 
                   ∑ 
                   
                     
                       n 
                       2 
                     
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                   
                     1 
                     
                       n 
                       2 
                     
                   
                    
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                         π 
                         2 
                       
                     
                   
                    
                   
                     
                       J 
                       
                         n 
                         2 
                       
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                     { 
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                       + 
                       
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                               - 
                               1 
                             
                             ] 
                           
                         
                          
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             n 
                             2 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                     } 
                   
                 
               
             
           
         
       
     
         [0177]    Similarly inspect L 2  for a summation over n 1  in two steps: 
         [0000]    
       
         
           
             
               
                 
                   ∑ 
                   
                     
                       
                         n 
                         1 
                       
                       = 
                       
                         - 
                         ∞ 
                       
                     
                     , 
                     
                       
                         n 
                         1 
                       
                       ≠ 
                       0 
                     
                   
                   ∞ 
                 
                  
                 
                   
                     
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 n 
                                 1 
                               
                             
                             ) 
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                     
                       n 
                       1 
                     
                   
                    
                   
                     
                       J 
                       
                         n 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         n 
                         1 
                       
                        
                       
                         φ 
                         ′ 
                       
                     
                   
                 
               
               == 
               
                 
                   
                     ∑ 
                     
                       
                         n 
                         1 
                       
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   n 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                       
                         n 
                         1 
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         
                           n 
                           1 
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       
                         n 
                         1 
                       
                       = 
                       
                         - 
                         ∞ 
                       
                     
                     
                       - 
                       1 
                     
                   
                    
                   
                     
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   n 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                       
                         n 
                         1 
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         
                           n 
                           1 
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                   
                 
               
             
             = 
             
               
                 
                   
                     ∑ 
                     
                       
                         n 
                         1 
                       
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   n 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                       
                         n 
                         1 
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         
                           n 
                           1 
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       
                         n 
                         1 
                       
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   n 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                             
                             - 
                             1 
                           
                           ] 
                         
                       
                       
                         n 
                         1 
                       
                     
                      
                     
                       
                         J 
                         
                           - 
                           
                             n 
                             1 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         
                           n 
                           1 
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                   
                 
               
               = 
               * 
             
           
         
       
     
         [0178]    Using J −n     1   (k 0 |z−r|)=(−1) n     1   J n     1   (k 0 |z−r|) 
         [0000]    
       
         
           
             *= 
             
               
                 
                   ∑ 
                   
                     
                       n 
                       1 
                     
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                   
                     
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 n 
                                 1 
                               
                             
                             ) 
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                     
                       n 
                       1 
                     
                   
                    
                   
                     
                       J 
                       
                         n 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         n 
                         1 
                       
                        
                       
                         φ 
                         ′ 
                       
                     
                   
                 
               
               + 
               
                 
                   
                     ∑ 
                     
                       
                         n 
                         1 
                       
                       = 
                       1 
                     
                   
                   ∞ 
                 
                  
                 
                   
                     
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   n 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                             
                             - 
                             1 
                           
                           ] 
                         
                       
                        
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           n 
                           1 
                         
                       
                     
                     
                       n 
                       1 
                     
                   
                    
                   
                     
                       J 
                       
                         n 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       
                         n 
                         1 
                       
                        
                       
                         φ 
                         ′ 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               L 
               2 
             
             = 
             
               N 
                
               
                   
               
                
               ξ 
                
               
                   
               
                
               
                 
                   J 
                   0 
                 
                  
                 
                   ( 
                   
                     
                       k 
                       0 
                     
                      
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                   
                   ) 
                 
               
                
               
                 
                   ∑ 
                   
                     
                       n 
                       1 
                     
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                   
                     
                        
                       
                          
                          
                         
                           π 
                           2 
                         
                       
                     
                     
                       n 
                       1 
                     
                   
                    
                   
                     
                       J 
                       
                         n 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                     { 
                     
                       
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     1 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                       + 
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     1 
                                   
                                    
                                   ξ 
                                 
                               
                               - 
                               1 
                             
                             ] 
                           
                         
                          
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             n 
                             1 
                           
                         
                          
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                     } 
                   
                 
               
             
           
         
       
     
         [0179]    Which is the complex conjugate of L 1  as it should be. 
         [0000]    
       
         
           
             
               L 
               1 
             
             = 
             
               N 
                
               
                   
               
                
               ξ 
                
               
                   
               
                
               
                 
                   J 
                   0 
                 
                  
                 
                   ( 
                   
                     
                       k 
                       0 
                     
                      
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                   
                   ) 
                 
               
                
               
                 
                   ∑ 
                   
                     
                       n 
                       2 
                     
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                   
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                           π 
                           2 
                         
                       
                     
                     
                       n 
                       2 
                     
                   
                    
                   
                     
                       J 
                       
                         n 
                         2 
                       
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                     { 
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                       + 
                       
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                               - 
                               1 
                             
                             ] 
                           
                         
                          
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             n 
                             2 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                     } 
                   
                 
               
             
           
         
       
     
         [0180]    Summing up the terms for n 1  and n 2 , i.e. (L 1 +L 2 ) and denoting the index by n: 
         [0000]    
       
         
           
             N 
              
             
                 
             
              
             ξ 
              
             
                 
             
              
             
               
                 J 
                 0 
               
                
               
                 ( 
                 
                   
                     k 
                     0 
                   
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                 ) 
               
             
             × 
             
               { 
               
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                         + 
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                                 - 
                                 1 
                               
                               ] 
                             
                           
                            
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             n 
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       
                          
                         
                            
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                         + 
                         
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                                 - 
                                 1 
                               
                               ] 
                             
                           
                            
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             n 
                           
                            
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               } 
             
           
         
       
     
         [0181]    Rearranging terms multiplying e −inφ′  and e inφ′  separately we get: 
         [0000]    
       
         
           
             = 
             
               
                 N 
                  
                 
                     
                 
                  
                 ξ 
                  
                 
                     
                 
                  
                 
                   
                     J 
                     0 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                     ) 
                   
                 
                  
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       { 
                       
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         - 
                         
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                             n 
                           
                            
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             
                               
                                 J 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             n 
                           
                         
                       
                       } 
                     
                      
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         n 
                          
                         
                             
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                   
                 
               
               + 
               
                 N 
                  
                 
                     
                 
                  
                 ξ 
                  
                 
                     
                 
                  
                 
                   
                     J 
                     0 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                     ) 
                   
                 
                  
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                     
                       { 
                       
                         
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         - 
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                             n 
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             
                               
                                 J 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             n 
                           
                         
                       
                       } 
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         n 
                          
                         
                             
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0182]    Now inspect the upper term: 
         [0000]    
       
         
           
             
               
                 { 
                 
                   
                     
                       
                          
                         
                            
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           n 
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                   
                   - 
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         n 
                          
                         
                           π 
                           2 
                         
                       
                     
                      
                     
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       n 
                     
                   
                 
                 } 
               
                
               
                  
                 
                   
                     - 
                      
                   
                    
                   
                       
                   
                    
                   n 
                    
                   
                       
                   
                    
                   
                     φ 
                     ′ 
                   
                 
               
             
             = 
             
               
                 
                   { 
                   
                     
                       1 
                       n 
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             n 
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             n 
                           
                         
                         ] 
                       
                     
                   
                   } 
                 
                  
                 
                    
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     n 
                      
                     
                         
                     
                      
                     
                       φ 
                       ′ 
                     
                   
                 
               
               = 
               
                 
                   { 
                   
                     
                       
                          
                         
                            
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             n 
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           1 
                           + 
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             n 
                           
                         
                         ] 
                       
                     
                   
                   } 
                 
                  
                 
                    
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     n 
                      
                     
                         
                     
                      
                     
                       φ 
                       ′ 
                     
                   
                 
               
             
           
         
       
     
         [0183]    And similarly for the lower term: 
         [0000]    
       
         
           
             
               
                 { 
                 
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                   
                   - 
                   
                     
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         n 
                          
                         
                           π 
                           2 
                         
                       
                     
                      
                     
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       n 
                     
                   
                 
                 } 
               
                
               
                  
                 
                    
                    
                   
                       
                   
                    
                   n 
                    
                   
                       
                   
                    
                   
                     φ 
                     ′ 
                   
                 
               
             
             = 
             
               
                 { 
                 
                   
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                           π 
                           2 
                         
                       
                     
                     n 
                   
                    
                   
                     
                       J 
                       n 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       
                         1 
                         + 
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           n 
                         
                       
                       ] 
                     
                   
                 
                 } 
               
                
               
                  
                 
                    
                    
                   
                       
                   
                    
                   n 
                    
                   
                       
                   
                    
                   
                     φ 
                     ′ 
                   
                 
               
             
           
         
       
     
         [0184]    This verifies that by adding L 1  and L 2 , only even terms contribute to the summation, i.e. 
         [0000]    
       
         
           
             
               
                 L 
                 1 
               
               + 
               
                 L 
                 2 
               
             
             = 
             
               
                 
                   N 
                    
                   
                       
                   
                    
                   ξ 
                    
                   
                       
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                       
                         { 
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   n 
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   n 
                                 
                               
                               ] 
                             
                           
                         
                         } 
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                               
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
                 + 
                 
                   
                     { 
                     
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                         n 
                       
                        
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           [ 
                           
                             1 
                             + 
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               n 
                             
                           
                           ] 
                         
                       
                     
                     } 
                   
                    
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       n 
                        
                       
                           
                       
                        
                       
                         φ 
                         ′ 
                       
                     
                   
                 
               
               == 
               
                 
                   2 
                    
                   N 
                    
                   
                       
                   
                    
                   ξ 
                    
                   
                       
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                     
                       ∑ 
                       
                         
                           n 
                           = 
                           1 
                         
                         , 
                         even 
                       
                       ∞ 
                     
                      
                     
                       
                         { 
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         } 
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                               
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
                 + 
                 
                   
                     { 
                     
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                         n 
                       
                        
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             n 
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                     
                     } 
                   
                    
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       n 
                        
                       
                           
                       
                        
                       
                         φ 
                         ′ 
                       
                     
                   
                 
               
             
           
         
       
     
         [0185]    We now Fourier transform 
         [0000]    
       
         
           
             
               ∫ 
               
                 
                   
                      
                     2 
                   
                    
                   
                     ( 
                     
                       z 
                       - 
                       r 
                     
                     ) 
                   
                 
                  
                 
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         k 
                         · 
                         
                           [ 
                           
                             z 
                             - 
                             r 
                           
                           ] 
                         
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         L 
                         1 
                       
                       + 
                       
                         L 
                         2 
                       
                     
                     ) 
                   
                 
               
             
             == 
             
               
                 ∫ 
                 
                   
                     
                        
                       2 
                     
                      
                     
                       ( 
                       
                         z 
                         - 
                         r 
                       
                       ) 
                     
                   
                    
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         k 
                         · 
                         
                           [ 
                           
                             z 
                             - 
                             r 
                           
                           ] 
                         
                       
                     
                   
                   × 
                   2 
                    
                   N 
                    
                   
                       
                   
                    
                   ξ 
                    
                   
                       
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                    
                   
                     
                       ∑ 
                       
                         
                           n 
                           = 
                           1 
                         
                         , 
                         even 
                       
                       ∞ 
                     
                      
                     
                       
                         { 
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         } 
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                               
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               + 
               
                 
                   { 
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                   
                   } 
                 
                  
                 
                    
                   
                      
                      
                     
                         
                     
                      
                     n 
                      
                     
                         
                     
                      
                     
                       φ 
                       ′ 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               exp 
                
               
                 { 
                 
                   
                     - 
                      
                   
                    
                   
                       
                   
                    
                   
                     k 
                     · 
                     
                       [ 
                       
                         z 
                         - 
                         r 
                       
                       ] 
                     
                   
                 
                 } 
               
             
             = 
             
               
                 exp 
                  
                 
                   { 
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     
                        
                       k 
                        
                     
                      
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     
                       ( 
                       α 
                       ) 
                     
                   
                   } 
                 
               
               = 
               
                 
                   
                     
                       [ 
                       
                         
                           ∑ 
                           
                             
                               n 
                               3 
                             
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           ∞ 
                         
                          
                         
                           
                              
                             
                               n 
                               3 
                             
                           
                            
                           
                             
                               J 
                               
                                 n 
                                 3 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             exp 
                              
                             
                               ( 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   3 
                                 
                                  
                                 α 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     * 
                   
                   == 
                   
                     
                       ∑ 
                       
                         
                           n 
                           3 
                         
                         = 
                         
                           - 
                           ∞ 
                         
                       
                       ∞ 
                     
                      
                     
                       
                          
                         
                           - 
                           
                             n 
                             3 
                           
                         
                       
                        
                       
                         
                           J 
                           n 
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               3 
                             
                              
                             α 
                           
                           ) 
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         
                           n 
                           3 
                         
                         = 
                         
                           - 
                           ∞ 
                         
                       
                       ∞ 
                     
                      
                     
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             3 
                           
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               
                                 n 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   
                                     φ 
                                     ′ 
                                   
                                   + 
                                   
                                     π 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             
                               n 
                               3 
                             
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               φ 
                             
                           
                            
                           
                             
                               J 
                               
                                 n 
                                 3 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             exp 
                              
                             
                               ( 
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     n 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         φ 
                                         ′ 
                                       
                                       + 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             
                               n 
                               3 
                             
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           
                             - 
                             1 
                           
                         
                          
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               φ 
                             
                           
                            
                           
                             
                               J 
                               
                                 n 
                                 3 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             exp 
                              
                             
                               ( 
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     n 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         φ 
                                         ′ 
                                       
                                       + 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                  
                                 k 
                                  
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 3 
                               
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   3 
                                 
                                  
                                 φ 
                               
                             
                              
                             
                               
                                 J 
                                 
                                   n 
                                   3 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                      
                                     k 
                                      
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               exp 
                                
                               
                                 ( 
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       n 
                                       3 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           ′ 
                                         
                                         + 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 3 
                               
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   3 
                                 
                                  
                                 φ 
                               
                             
                              
                             
                               
                                 J 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                      
                                     k 
                                      
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               exp 
                                
                               
                                 ( 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       n 
                                       3 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           ′ 
                                         
                                         + 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           * 
                           
                             
 
                           
                            
                           
                             
                               J 
                               
                                 - 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                         = 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 
                                   n 
                                   3 
                                 
                               
                                
                               
                                 
                                   J 
                                   
                                     n 
                                     3 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                        
                                       k 
                                        
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                             
                             *= 
                             
                               
                                 
                                   J 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                        
                                       k 
                                        
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     
                                       n 
                                       3 
                                     
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       
                                         n 
                                         3 
                                       
                                        
                                       φ 
                                     
                                   
                                    
                                   
                                     
                                       J 
                                       
                                         n 
                                         3 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                            
                                           k 
                                            
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     exp 
                                      
                                     
                                       ( 
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             n 
                                             3 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 φ 
                                                 ′ 
                                               
                                               + 
                                               
                                                 π 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     
                                       n 
                                       3 
                                     
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         n 
                                         3 
                                       
                                        
                                       φ 
                                     
                                   
                                    
                                   
                                     
                                       J 
                                       
                                         n 
                                         3 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                            
                                           k 
                                            
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     exp 
                                      
                                     
                                       ( 
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             n 
                                             3 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 φ 
                                                 ′ 
                                               
                                               + 
                                               
                                                 π 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                     
                                       n 
                                       3 
                                     
                                   
                                 
                               
                             
                           
                           = 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0186]    Denoting F for compact notation: 
         [0000]    
       
         
           
             F 
             = 
             
               
                 
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                            
                           k 
                            
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         
                           n 
                           3 
                         
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                       { 
                       
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               φ 
                             
                           
                            
                           
                             
                               J 
                               
                                 n 
                                 3 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                         + 
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               φ 
                             
                           
                            
                           
                             
                               J 
                               
                                 n 
                                 3 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     3 
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             
                               n 
                               3 
                             
                           
                            
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                 
                                 3 
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
                  
                 
                   
 
                 
                  
                 
                     
                 
                 ⇒ 
                 
                   
 
                 
                  
                 
                   ∫ 
                   
                     
                       
                          
                         2 
                       
                        
                       
                         ( 
                         
                           z 
                           - 
                           r 
                         
                         ) 
                       
                     
                      
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             k 
                             · 
                             
                               [ 
                               
                                 z 
                                 - 
                                 r 
                               
                               ] 
                             
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             L 
                             1 
                           
                           + 
                           
                             L 
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     ∫ 
                     0 
                     ∞ 
                   
                    
                   
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                      
                     
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         ∫ 
                         0 
                         
                           2 
                            
                           π 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           
                             
                               φ 
                               ′ 
                             
                              
                             
                               ( 
                               
                                 
                                   L 
                                   1 
                                 
                                 + 
                                 
                                   L 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                          
                         F 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                      
                     
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                        
                       
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           ∫ 
                           0 
                           
                             2 
                              
                             π 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                              
                             
                               φ 
                               ′ 
                             
                           
                           × 
                           2 
                            
                           N 
                            
                           
                               
                           
                            
                           ξ 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 , 
                                 even 
                               
                               ∞ 
                             
                              
                             
                               
                                 { 
                                 
                                   
                                     
                                        
                                       
                                          
                                          
                                         
                                           π 
                                           2 
                                         
                                       
                                     
                                     n 
                                   
                                    
                                   
                                     
                                       J 
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           0 
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                           π 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                              
                                              
                                             
                                                 
                                             
                                              
                                             n 
                                              
                                             
                                                 
                                             
                                              
                                             ξ 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 } 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   
                                     φ 
                                     ′ 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   + 
                   
                     
                       { 
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                           n 
                         
                          
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         n 
                          
                         
                             
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                     × 
                     
                       { 
                       
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                  
                                 k 
                                  
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 3 
                               
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             [ 
                             
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       3 
                                     
                                      
                                     φ 
                                   
                                 
                                  
                                 
                                   
                                     J 
                                     
                                       n 
                                       3 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                          
                                         k 
                                          
                                       
                                        
                                       
                                          
                                         
                                           z 
                                           - 
                                           r 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       3 
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       3 
                                     
                                      
                                     
                                       φ 
                                       ′ 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       3 
                                     
                                      
                                     φ 
                                   
                                 
                                  
                                 
                                   
                                     J 
                                     
                                       n 
                                       3 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                          
                                         k 
                                          
                                       
                                        
                                       
                                          
                                         
                                           z 
                                           - 
                                           r 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         
                                           n 
                                           3 
                                         
                                          
                                         
                                           π 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   
                                     n 
                                     3 
                                   
                                 
                                  
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         n 
                                          
                                         
                                             
                                         
                                       
                                       3 
                                     
                                      
                                     
                                       φ 
                                       ′ 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
             
           
         
       
     
         [0187]    Note that: 
         [0000]      ∫ 0   2π   dφ′e   inφ′ =∫ 0   2π   dφ′e   −inφ′ =0
 
         [0000]      ∫ 0   2π   dφ′e   inφ′   e   in     3     φ′ =∫ 0   2π   dφ′e   −inφ′   e   −in     3     φ′ =0
 
         [0000]      ∫ 0   2π   dφ′e   inφ′   e   −in     3     φ′ =∫ 0   2π   dφ′e   −inφ′   e   in     3     φ′ =2πδ n,n     3    
 
         [0188]    Therefore the only surviving terms are: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
                  
                 
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         
                           φ 
                           ′ 
                         
                          
                         
                           ( 
                           
                             
                               L 
                               1 
                             
                             + 
                             
                               L 
                               2 
                             
                           
                           ) 
                         
                       
                     
                      
                     F 
                   
                 
               
             
             = 
             
               
                 
                   ∫ 
                   0 
                   ∞ 
                 
                  
                 
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                    
                   
                      
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                   
                   × 
                   2 
                    
                   N 
                    
                   
                       
                   
                    
                   ξ 
                    
                   
                       
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           k 
                           0 
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                   × 
                   
                     { 
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             even 
                           
                           ∞ 
                         
                          
                         
                           
                             { 
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                 n 
                               
                                
                               
                                 
                                   J 
                                   n 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             } 
                           
                            
                           
                             { 
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     3 
                                   
                                   = 
                                   1 
                                 
                                 ∞ 
                               
                                
                               
                                 [ 
                                 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       
                                         n 
                                         3 
                                       
                                        
                                       φ 
                                     
                                   
                                    
                                   
                                     
                                       J 
                                       
                                         n 
                                         3 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                            
                                           k 
                                            
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         n 
                                         3 
                                       
                                        
                                       
                                         π 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             } 
                           
                            
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           
                             δ 
                             
                               n 
                               , 
                               
                                 n 
                                 3 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             even 
                           
                           ∞ 
                         
                          
                         
                           
                             { 
                             
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                 n 
                               
                                
                               
                                 
                                   J 
                                   n 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             } 
                           
                            
                           
                             { 
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     3 
                                   
                                   = 
                                   1 
                                 
                                 ∞ 
                               
                                
                               
                                 [ 
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         n 
                                         3 
                                       
                                        
                                       φ 
                                     
                                   
                                    
                                   
                                     
                                       J 
                                       
                                         n 
                                         3 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                            
                                           k 
                                            
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       
                                          
                                         
                                            
                                            
                                           
                                               
                                           
                                            
                                           
                                             n 
                                             3 
                                           
                                            
                                           
                                             π 
                                             2 
                                           
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     
                                       n 
                                       3 
                                     
                                   
                                 
                                 ] 
                               
                             
                             } 
                           
                            
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           
                             δ 
                             
                               n 
                               , 
                               
                                 n 
                                 3 
                               
                             
                           
                         
                       
                     
                     } 
                   
                 
               
               = 
               
                 4 
                  
                 π 
                  
                 
                     
                 
                  
                 N 
                  
                 
                     
                 
                  
                 ξ 
                  
                 
                   
                     ∫ 
                     0 
                     ∞ 
                   
                    
                   
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                      
                     
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                      
                     
                       
                         J 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           even 
                         
                         ∞ 
                       
                        
                       
                         { 
                         
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                               n 
                             
                              
                             
                               
                                 J 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       φ 
                                     
                                   
                                    
                                   
                                     
                                       J 
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                            
                                           k 
                                            
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                         π 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               
                                  
                                 
                                    
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                               n 
                             
                              
                             
                               
                                 J 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       φ 
                                     
                                   
                                    
                                   
                                     
                                       J 
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                            
                                           k 
                                            
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                         π 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
             
           
         
       
     
         [0189]    We note that the above two terms are complex conjugates of each other, so according to footnote 12    
         [0000]    
       
         
           
             
               ∫ 
               
                 
                   
                      
                     2 
                   
                    
                   
                     ( 
                     
                       z 
                       - 
                       r 
                     
                     ) 
                   
                 
                  
                 
                   ( 
                   
                     
                       L 
                       1 
                     
                     + 
                     
                       L 
                       2 
                     
                   
                   ) 
                 
                  
                 F 
               
             
             = 
             
               
                 4 
                  
                 π 
                  
                 
                     
                 
                  
                 N 
                  
                 
                     
                 
                  
                 ξ 
                  
                 
                   
                     ∫ 
                     0 
                     ∞ 
                   
                    
                   
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                      
                     
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                      
                     
                       
                         J 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                     × 
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           even 
                         
                         ∞ 
                       
                        
                       
                         2 
                          
                         
                             
                         
                          
                         Re 
                          
                         
                           { 
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                               n 
                             
                              
                             
                               
                                 J 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       φ 
                                     
                                   
                                    
                                   
                                     
                                       J 
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                            
                                           k 
                                            
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                         π 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
               = 
               
                 
                   4 
                    
                   π 
                    
                   
                       
                   
                    
                   N 
                    
                   
                       
                   
                    
                   ξ 
                    
                   
                     
                       ∑ 
                       
                         
                           n 
                           = 
                           1 
                         
                         , 
                         even 
                       
                       ∞ 
                     
                      
                     
                       2 
                        
                       
                           
                       
                        
                       Re 
                        
                       
                         { 
                         
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                             n 
                           
                            
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               φ 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                 π 
                                 2 
                               
                             
                           
                         
                         } 
                       
                        
                       
                         I 
                         
                           0 
                           , 
                           n 
                           , 
                           n 
                         
                       
                     
                   
                 
                 = 
                 * 
               
             
           
         
       
     
         [0000]      12  C=A+iB; C+C*=2A=2Re(C) 
         [0190]    Inspect: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         n 
                          
                         
                             
                         
                          
                         φ 
                       
                     
                      
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         n 
                          
                         
                           π 
                           2 
                         
                       
                     
                   
                   = 
                     
                    
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                               
                           
                            
                           π 
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         n 
                          
                         
                             
                         
                          
                         φ 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                         n 
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                               
                           
                            
                           π 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           n 
                            
                           
                               
                           
                            
                           φ 
                         
                       
                     
                     - 
                     
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                         n 
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                              
                             
                               ( 
                               
                                 π 
                                 + 
                                 ξ 
                               
                               ) 
                             
                           
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           n 
                            
                           
                               
                           
                            
                           φ 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                       n 
                     
                      
                     
                       [ 
                       
                         
                            
                           
                              
                              
                             
                               ( 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   φ 
                                 
                                 - 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                 
                               
                               ) 
                             
                           
                         
                         - 
                         
                            
                           
                              
                              
                             
                               ( 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   φ 
                                 
                                 - 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                 
                                 - 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   ξ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                           π 
                           2 
                         
                       
                     
                     n 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     [ 
                     
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                   - 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                                
                                
                               
                                   
                               
                                
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                   - 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                 
                                 ) 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                   - 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                   - 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                                
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       φ 
                                     
                                     - 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                     
                                     - 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
         [0191]    Taking the Real part (note that 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                         π 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         cos 
                          
                         
                           ( 
                           
                             - 
                             
                               π 
                               2 
                             
                           
                           ) 
                         
                       
                       + 
                       
                          
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               - 
                               
                                 π 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       - 
                        
                     
                   
                 
                 ) 
               
                
               
                 : 
               
                
               
                   
               
                
               Re 
                
               
                 { 
                 
                   
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                           π 
                           2 
                         
                       
                     
                     n 
                   
                    
                   
                     [ 
                     
                       
                         cos 
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             
                               n 
                                
                               
                                   
                               
                                
                               φ 
                             
                             - 
                             
                               n 
                                
                               
                                   
                               
                                
                               π 
                             
                           
                           ) 
                         
                       
                       + 
                       
                          
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 φ 
                               
                               - 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               n 
                                
                               
                                   
                               
                                
                               φ 
                             
                             - 
                             
                               n 
                                
                               
                                   
                               
                                
                               π 
                             
                             - 
                             
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                           
                           ) 
                         
                       
                       - 
                       
                          
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 φ 
                               
                               - 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               - 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
             = 
             
               
                 
                   1 
                   n 
                 
                  
                 Re 
                  
                 
                   { 
                   
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         ( 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             φ 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             φ 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       cos 
                        
                       
                         ( 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             φ 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             π 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             ξ 
                           
                         
                         ) 
                       
                     
                     - 
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             φ 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             π 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             ξ 
                           
                         
                         ) 
                       
                     
                   
                   } 
                 
               
               = 
               
                 
                   1 
                   n 
                 
                  
                 
                   { 
                   
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             φ 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             φ 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             π 
                           
                           - 
                           
                             n 
                              
                             
                                 
                             
                              
                             ξ 
                           
                         
                         ) 
                       
                     
                   
                   } 
                 
               
             
           
         
       
     
         [0192]    We finally arrive at: 
         [0000]    
       
         
           
             
               ∫ 
               
                 
                   
                      
                     2 
                   
                    
                   
                     ( 
                     
                       z 
                       - 
                       r 
                     
                     ) 
                   
                 
                  
                 
                    
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     
                       k 
                       · 
                       
                         [ 
                         
                           z 
                           - 
                           r 
                         
                         ] 
                       
                     
                   
                 
                  
                 
                   { 
                   
                     
                       L 
                       1 
                     
                     + 
                     
                       L 
                       2 
                     
                   
                   } 
                 
               
             
             = 
             
               8 
                
               π 
                
               
                   
               
                
               N 
                
               
                   
               
                
               ξ 
                
               
                 
                   ∑ 
                   
                     
                       n 
                       = 
                       1 
                     
                     , 
                     even 
                   
                   ∞ 
                 
                  
                 
                   
                     
                       1 
                       n 
                     
                      
                     
                       [ 
                       
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 φ 
                               
                               - 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 φ 
                               
                               - 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               - 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                    
                   
                     I 
                     
                       0 
                       , 
                       n 
                       , 
                       n 
                     
                   
                 
               
             
           
         
       
     
         [0193]    Using the identity in footnote 13    
         [0000]    
       
         
           
             
               ∫ 
               
                 
                   
                      
                     2 
                   
                    
                   
                     ( 
                     
                       z 
                       - 
                       r 
                     
                     ) 
                   
                 
                  
                 
                    
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     
                       k 
                       · 
                       
                         [ 
                         
                           z 
                           - 
                           r 
                         
                         ] 
                       
                     
                   
                 
                  
                 
                   { 
                   
                     
                       L 
                       1 
                     
                     + 
                     
                       L 
                       2 
                     
                   
                   } 
                 
               
             
             = 
             
               8 
                
               π 
                
               
                   
               
                
               N 
                
               
                   
               
                
               ξ 
                
               
                 
                   
                     ∑ 
                     
                       
                         n 
                         = 
                         1 
                       
                       , 
                       even 
                     
                   
                   ∞ 
                 
                  
                 
                   
                     
                       1 
                       n 
                     
                      
                     
                       [ 
                       
                         
                           sin 
                            
                           
                             ( 
                             
                               n 
                                
                               
                                   
                               
                                
                               φ 
                             
                             ) 
                           
                         
                         - 
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 φ 
                               
                               - 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                    
                   
                     I 
                     
                       0 
                       , 
                       n 
                       , 
                       n 
                     
                   
                 
               
             
           
         
       
     
         [0194]    For ξ=π we get: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 sin 
                  
                 
                   ( 
                   
                     n 
                      
                     
                       ( 
                       
                         π 
                         - 
                         φ 
                       
                       ) 
                     
                   
                   ) 
                 
               
               = 
               
                 - 
                 
                   sin 
                    
                   
                     ( 
                     
                       
                         n 
                          
                         
                             
                         
                          
                         φ 
                       
                       - 
                       
                         n 
                          
                         
                             
                         
                          
                         π 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   sin 
                    
                   
                     ( 
                     
                       
                         n 
                          
                         
                             
                         
                          
                         φ 
                       
                       - 
                       
                         n 
                          
                         
                             
                         
                          
                         π 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   sin 
                    
                   
                     ( 
                     
                       n 
                        
                       
                           
                       
                        
                       φ 
                     
                     ) 
                   
                 
               
               ; 
               
                 n 
                 = 
                 
                   even 
                    
                   
                     
 
                   
                    
                   
                       
                   
                   ⇒ 
                 
               
             
           
         
       
     
         [0195]    http://en.wikipedia.org/wiki/Angle_addition_formula#Angle_sum_and_difference_identities 
         [0196]    sin(a−b)=sin a cos b−cos a sin b 
         [0197]               sin(a−nπ)=sin a cos nπ−cos a sin nπ=sin a; for n=even 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         
                           
                              
                             2 
                           
                            
                           
                             ( 
                             
                               z 
                               - 
                               r 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               · 
                               
                                 [ 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ] 
                               
                             
                           
                         
                          
                         
                           { 
                           
                             
                               L 
                               1 
                             
                             + 
                             
                               L 
                               2 
                             
                           
                           } 
                         
                       
                     
                      
                     
                       | 
                       
                         ξ 
                         = 
                         π 
                       
                     
                   
                   = 
                     
                    
                   
                     8 
                      
                     π 
                      
                     
                         
                     
                      
                     N 
                      
                     
                         
                     
                      
                     ξ 
                      
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           even 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             1 
                             n 
                           
                            
                           
                             [ 
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           I 
                           
                             0 
                             , 
                             n 
                             , 
                             n 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   0 
                 
               
             
           
         
       
     
         [0198]    We now look at: ∫d 2 (z−r)e −ik·[z−r] {L 0 } 
         [0000]    
       
         
           
             
               ∫ 
               
                 
                   
                      
                     2 
                   
                    
                   
                     ( 
                     
                       z 
                       - 
                       r 
                     
                     ) 
                   
                 
                  
                 
                    
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     
                       k 
                       · 
                       
                         [ 
                         
                           z 
                           - 
                           r 
                         
                         ] 
                       
                     
                   
                 
                  
                 
                   { 
                   
                     L 
                     0 
                   
                   } 
                 
               
             
             = 
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
                  
                 
                     
                 
                  
                 
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         ′ 
                       
                     
                      
                     
                       L 
                       0 
                     
                      
                     F 
                   
                 
               
             
           
         
       
     
         [0199]    Where as before 
         [0000]    
       
         
           
             F 
             = 
             
               
                 
                   J 
                   0 
                 
                  
                 
                   ( 
                   
                     
                        
                       k 
                        
                     
                      
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                   
                   ) 
                 
               
               + 
               
                 
                   ∑ 
                   
                     
                       n 
                       3 
                     
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                     
                 
                  
                 
                   { 
                   
                     
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           φ 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             3 
                           
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                     + 
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           φ 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             3 
                           
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                         
                         
                           n 
                           3 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                   } 
                 
               
             
           
         
       
     
         [0000]    and L 0 =N{ξ 2 J 0 (k 0 |z−r|)J 0 (k 0 |z−r|). We get 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
                  
                 
                     
                 
                  
                 
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         ′ 
                       
                     
                      
                     
                       L 
                       0 
                     
                      
                     F 
                   
                 
               
             
             = 
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
                  
                 
                     
                 
                  
                 
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         ′ 
                       
                     
                      
                     N 
                      
                     
                       { 
                       
                         
                           
                             ξ 
                             2 
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               J 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                    
                                   π 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                    
                                   
                                     φ 
                                     ′ 
                                   
                                 
                                  
                                 N 
                                  
                                 
                                   { 
                                   
                                     
                                       
                                         ξ 
                                         2 
                                       
                                        
                                       
                                         
                                           J 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           J 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                       × 
                                       
                                         
                                           ∑ 
                                           
                                             
                                               n 
                                               3 
                                             
                                             = 
                                             1 
                                           
                                           ∞ 
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           { 
                                           
                                             
                                               
                                                  
                                                 
                                                    
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     n 
                                                     3 
                                                   
                                                    
                                                   φ 
                                                 
                                               
                                                
                                               
                                                 
                                                   J 
                                                   
                                                     n 
                                                     3 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                        
                                                       k 
                                                        
                                                     
                                                      
                                                     
                                                        
                                                       
                                                         z 
                                                         - 
                                                         r 
                                                       
                                                        
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                                
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                      
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     n 
                                                     3 
                                                   
                                                    
                                                   
                                                     π 
                                                     2 
                                                   
                                                 
                                               
                                                
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                      
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     n 
                                                     3 
                                                   
                                                    
                                                   
                                                     φ 
                                                     ′ 
                                                   
                                                 
                                               
                                             
                                             + 
                                             
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                      
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     n 
                                                     3 
                                                   
                                                    
                                                   φ 
                                                 
                                               
                                                
                                               
                                                 
                                                   J 
                                                   
                                                     n 
                                                     3 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                        
                                                       k 
                                                        
                                                     
                                                      
                                                     
                                                        
                                                       
                                                         z 
                                                         - 
                                                         r 
                                                       
                                                        
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                                
                                               
                                                 
                                                   
                                                      
                                                     
                                                        
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         n 
                                                         3 
                                                       
                                                        
                                                       
                                                         π 
                                                         2 
                                                       
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       - 
                                                       1 
                                                     
                                                     ) 
                                                   
                                                 
                                                 
                                                   n 
                                                   3 
                                                 
                                               
                                                
                                               
                                                  
                                                 
                                                    
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     n 
                                                     3 
                                                   
                                                    
                                                   
                                                     φ 
                                                     ′ 
                                                   
                                                 
                                               
                                             
                                           
                                           } 
                                         
                                       
                                     
                                     == 
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         N 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ξ 
                                           2 
                                         
                                          
                                         
                                           I 
                                           
                                             0 
                                             , 
                                             0 
                                             , 
                                             0 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           ∫ 
                                           0 
                                           ∞ 
                                         
                                          
                                         
                                           
                                              
                                             
                                               z 
                                               - 
                                               r 
                                             
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                              
                                             
                                                
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                                
                                             
                                           
                                            
                                           
                                             
                                               ∫ 
                                               0 
                                               
                                                 2 
                                                  
                                                 π 
                                               
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               
                                                  
                                                 
                                                   φ 
                                                   ′ 
                                                 
                                               
                                                
                                               N 
                                                
                                               
                                                 { 
                                                 
                                                   
                                                     
                                                       
                                                         ξ 
                                                         2 
                                                       
                                                        
                                                       
                                                         
                                                           J 
                                                           0 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             k 
                                                             0 
                                                             
                                                              
                                                             
                                                              
                                                             
                                                             z 
                                                             - 
                                                             r 
                                                             
                                                              
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                        
                                                       
                                                         
                                                           J 
                                                           0 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             k 
                                                             0 
                                                             
                                                              
                                                             
                                                              
                                                             
                                                             z 
                                                             - 
                                                             r 
                                                             
                                                              
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       × 
                                                       
                                                         
                                                           ∑ 
                                                           
                                                             
                                                             n 
                                                             3 
                                                             
                                                             = 
                                                             1 
                                                           
                                                           ∞ 
                                                         
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         
                                                           
                                                             { 
                                                             
                                                             
                                                             
                                                              
                                                             
                                                              
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             n 
                                                             3 
                                                             
                                                              
                                                             φ 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             J 
                                                             
                                                             n 
                                                             3 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                              
                                                             k 
                                                              
                                                             
                                                              
                                                             
                                                              
                                                             
                                                             z 
                                                             - 
                                                             r 
                                                             
                                                              
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                              
                                                             
                                                              
                                                             
                                                             
                                                             - 
                                                              
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             n 
                                                             3 
                                                             
                                                              
                                                             
                                                             π 
                                                             2 
                                                             
                                                             
                                                             
                                                              
                                                             
                                                              
                                                             
                                                             
                                                             - 
                                                              
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             n 
                                                             3 
                                                             
                                                              
                                                             
                                                             φ 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             - 
                                                              
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             n 
                                                             3 
                                                             
                                                              
                                                             φ 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             J 
                                                             
                                                             n 
                                                             3 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                              
                                                             k 
                                                              
                                                             
                                                              
                                                             
                                                              
                                                             
                                                             z 
                                                             - 
                                                             r 
                                                             
                                                              
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             
                                                              
                                                             
                                                              
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             n 
                                                             3 
                                                             
                                                              
                                                             
                                                             π 
                                                             2 
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             - 
                                                             1 
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             n 
                                                             3 
                                                             
                                                             
                                                              
                                                             
                                                              
                                                             
                                                              
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             n 
                                                             3 
                                                             
                                                              
                                                             
                                                             φ 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             } 
                                                           
                                                            
                                                           
                                                             
                                                             ∫ 
                                                             0 
                                                             ∞ 
                                                             
                                                              
                                                             
                                                             
                                                              
                                                             
                                                             z 
                                                             - 
                                                             r 
                                                             
                                                              
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                              
                                                             
                                                              
                                                             
                                                             z 
                                                             - 
                                                             r 
                                                             
                                                              
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             ∫ 
                                                             0 
                                                             
                                                             2 
                                                              
                                                             π 
                                                             
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                              
                                                             
                                                             φ 
                                                             ′ 
                                                             
                                                             
                                                              
                                                             
                                                             L 
                                                             0 
                                                             
                                                              
                                                             F 
                                                             
                                                             
                                                             
                                                           
                                                         
                                                       
                                                     
                                                     = 
                                                     
                                                       2 
                                                        
                                                       π 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       N 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         ξ 
                                                         2 
                                                       
                                                        
                                                       
                                                         I 
                                                         
                                                           0 
                                                           , 
                                                           0 
                                                           , 
                                                           0 
                                                         
                                                       
                                                     
                                                   
                                                   ; 
                                                   
                                                     as 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       
                                                         ∫ 
                                                         0 
                                                         
                                                           2 
                                                            
                                                           π 
                                                         
                                                       
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         
                                                            
                                                           
                                                             φ 
                                                             ′ 
                                                           
                                                         
                                                          
                                                         
                                                            
                                                           
                                                             
                                                              
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             n 
                                                             3 
                                                             
                                                              
                                                             
                                                             φ 
                                                             ′ 
                                                             
                                                             
                                                             = 
                                                             0 
                                                           
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0200]    Finally we look at: 
         [0000]    
       
         
           
             
                 
             
              
             
               ∫ 
               
                 
                   
                      
                     2 
                   
                    
                   
                     ( 
                     
                       z 
                       - 
                       r 
                     
                     ) 
                   
                 
                  
                 
                    
                   
                     
                       - 
                        
                     
                      
                     
                         
                     
                      
                     
                       k 
                       · 
                       
                         [ 
                         
                           z 
                           - 
                           r 
                         
                         ] 
                       
                     
                   
                 
                  
                 
                   { 
                   
                     L 
                     3 
                   
                   } 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 ∫ 
                 
                   
                     
                        
                       2 
                     
                      
                     
                       ( 
                       
                         z 
                         - 
                         r 
                       
                       ) 
                     
                   
                    
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         k 
                         · 
                         
                           [ 
                           
                             z 
                             - 
                             r 
                           
                           ] 
                         
                       
                     
                   
                    
                   
                     { 
                     
                       L 
                       3 
                     
                     } 
                   
                 
               
               = 
               
                 
                   ∫ 
                   0 
                   ∞ 
                 
                  
                 
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                    
                   
                       
                   
                    
                   
                      
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                   
                    
                   
                     
                       ∫ 
                       0 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           φ 
                           ′ 
                         
                       
                        
                       
                         L 
                         3 
                       
                        
                       F 
                     
                   
                 
               
             
           
         
       
       
         
           
             F 
             = 
             
               
                 
                   J 
                   0 
                 
                  
                 
                   ( 
                   
                     
                        
                       k 
                        
                     
                      
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                   
                   ) 
                 
               
               + 
               
                 
                   ∑ 
                   
                     
                       n 
                       3 
                     
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                     
                 
                  
                 
                   { 
                   
                     
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           φ 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             3 
                           
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           
                             π 
                             2 
                           
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                     + 
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           φ 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             3 
                           
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                         
                         
                           n 
                           3 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             3 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                   } 
                 
               
             
           
         
       
       
         
           
             
               L 
               3 
             
             = 
             
               
                 N 
                  
                 
                   [ 
                   
                     
                       ∑ 
                       
                         
                           
                             n 
                             2 
                           
                           = 
                           
                             - 
                             ∞ 
                           
                         
                         , 
                         
                           
                             n 
                             2 
                           
                           ≠ 
                           0 
                         
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                   ] 
                 
               
               × 
               
                   
                 
                   [ 
                   
                     
                       ∑ 
                       
                         
                           
                             n 
                             1 
                           
                           = 
                           
                             - 
                             ∞ 
                           
                         
                         , 
                         
                           
                             n 
                             1 
                           
                           ≠ 
                           0 
                         
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     1 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           n 
                           1 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             1 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0201]    As before, we now partition the summation into two parts: 
         [0000]    
       
         
           
             
               L 
               3 
             
             = 
             
               N 
               × 
               
                 [ 
                 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         
                           - 
                           ∞ 
                         
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
                 ] 
               
               × 
               
                   
                 
                   [ 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             1 
                           
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             
                                
                               
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         n 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       1 
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                           
                             n 
                             1 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             n 
                             1 
                           
                           = 
                           
                             - 
                             ∞ 
                           
                         
                         
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             
                                
                               
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         n 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       1 
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                           
                             n 
                             1 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0202]    Again using: 
         [0000]        J   −n (| k∥z−r |)=(−1) n   J   n (| k∥z−r |)
 
         [0203]    We get: 
         [0000]    
       
         
           
             
               L 
               3 
             
             = 
             
               N 
               × 
               
                 [ 
                 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         
                           - 
                           
                             n 
                             2 
                           
                         
                       
                        
                       
                         
                           J 
                           
                             n 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
                 ] 
               
               × 
               
                   
                 
                   [ 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             1 
                           
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             
                                
                               
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         n 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       1 
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                           
                             n 
                             1 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             n 
                             1 
                           
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             
                                
                               
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       1 
                                     
                                      
                                     ξ 
                                   
                                 
                               
                               ] 
                             
                           
                           
                             - 
                             
                               n 
                               1 
                             
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             n 
                             1 
                           
                         
                          
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0204]    We first look at: 
         [0000]    
       
         
           
             
               ∫ 
               0 
               ∞ 
             
              
             
               
                  
                 
                   z 
                   - 
                   r 
                 
                  
               
                
               
                   
               
                
               
                  
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
               
                
               
                 
                   ∫ 
                   0 
                   
                     2 
                      
                     π 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                      
                     
                       φ 
                       ′ 
                     
                   
                    
                   
                     L 
                     3 
                   
                    
                   
                     
                       J 
                       0 
                     
                      
                     
                       ( 
                       
                         
                            
                           k 
                            
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0205]    Recalling that: 
         [0000]      ∫ 0   2π   dφ′e   inφ′ =∫ 0   2π   dφ′e   −inφ′ =0
 
         [0000]      ∫ 0   2π   dφ′e   in     1     φ′   e   in     2     φ′ =∫ 0   2π   dφ′e   −in     1     φ′   e   −in     2     φ′ =0
 
         [0000]      ∫ 0   2π   dφ′e   in     1     φ′   e   −in     2     φ′ =∫ 0   2π   dφ′e   −in     1     φ′   e   in     2     φ′ =2πδ n     1     ,n     2    
 
         [0206]    The only terms remaining are: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
                  
                 
                     
                 
                  
                 
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         ′ 
                       
                     
                      
                     
                       L 
                       3 
                     
                      
                     
                       
                         J 
                         0 
                       
                        
                       
                         ( 
                         
                           
                              
                             k 
                              
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             = 
             
               
                 
                   
                     ∫ 
                     0 
                     ∞ 
                   
                    
                   
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                      
                     
                         
                     
                      
                     
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                      
                     
                       
                         ∫ 
                         0 
                         
                           2 
                            
                           π 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           
                             φ 
                             ′ 
                           
                         
                          
                         
                           
                             J 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                  
                                 k 
                                  
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         N 
                         × 
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                       
                                          
                                          
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           n 
                                           2 
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               
                                 n 
                                 2 
                               
                             
                              
                             
                               
                                 J 
                                 
                                   n 
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                                  
                                 
                                   φ 
                                   ′ 
                                 
                               
                             
                             × 
                             
                               
                                 
                                    
                                   
                                     
                                        
                                        
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             n 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         
                                           n 
                                           1 
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               
                                 n 
                                 1 
                               
                             
                              
                             
                               
                                 J 
                                 
                                   n 
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   φ 
                                   ′ 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       
                         n 
                         1 
                       
                       , 
                       
                         
                           n 
                           2 
                         
                         = 
                         1 
                       
                     
                     ∞ 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         
                            
                           
                             
                               - 
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                       
                         - 
                         
                           n 
                           2 
                         
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         n 
                         2 
                       
                     
                      
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         
                           n 
                           2 
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                     × 
                     
                       
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                       
                         - 
                         
                           n 
                           1 
                         
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         n 
                         1 
                       
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         
                           n 
                           1 
                         
                          
                         
                           φ 
                           ′ 
                         
                       
                     
                   
                 
               
               = 
               
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   N 
                    
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                      
                     
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                        
                       
                           
                       
                        
                       
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                        
                       
                         
                           J 
                           0 
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                                
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                       × 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                     
                                        
                                        
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                   
                                 
                                 ] 
                               
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                           × 
                           
                             
                               
                                  
                                 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                   
                                 
                                 ] 
                               
                             
                             n 
                           
                            
                           
                             
                               J 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     ∞ 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         
                            
                           
                             
                               - 
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                       
                         - 
                         n 
                       
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       n 
                     
                     × 
                     
                       
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                           
                           ] 
                         
                       
                       
                         - 
                         n 
                       
                     
                      
                     
                       
                         J 
                         n 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       n 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               I 
               
                 
                   n 
                   1 
                 
                 , 
                 
                   n 
                   2 
                 
                 , 
                 
                   n 
                   3 
                 
               
             
             = 
             
               
                 
                   I 
                   
                     
                       n 
                       1 
                     
                     , 
                     
                       
                         n 
                         2 
                       
                        
                       
                         n 
                         3 
                       
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       k 
                       0 
                     
                     , 
                     
                        
                       k 
                        
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     ∫ 
                     0 
                     ∞ 
                   
                    
                   
                     
                        
                       
                         z 
                         - 
                         r 
                       
                        
                     
                      
                     
                         
                     
                      
                     
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         J 
                         
                           n 
                           3 
                         
                       
                        
                       
                         ( 
                         
                           
                              
                             k 
                              
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     2 
                      
                     π 
                      
                     
                         
                     
                      
                     N 
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                       
                                          
                                          
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               n 
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                        
                                        
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               n 
                             
                           
                           + 
                           
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                       
                                          
                                          
                                         
                                           ( 
                                           
                                             1 
                                             - 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               
                                 - 
                                 n 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                        
                                        
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               
                                 - 
                                 n 
                               
                             
                              
                             
                               I 
                               
                                 n 
                                 , 
                                 n 
                                 , 
                                 0 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       N 
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                                 n 
                               
                                
                               
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                                 n 
                               
                             
                             + 
                             
                               
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                                 
                                   - 
                                   n 
                                 
                               
                                
                               
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                                 
                                   - 
                                   n 
                                 
                               
                                
                               
                                 I 
                                 
                                   n 
                                   , 
                                   n 
                                   , 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         N 
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 
                                   2 
                                   
                                     n 
                                     2 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         ξ 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               I 
                               
                                 0 
                                 , 
                                 n 
                                 , 
                                 n 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   ξ 
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           
                             2 
                             - 
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                             - 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                             
                           
                           = 
                           
                             
                               2 
                               - 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       n 
                                     
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         - 
                                         n 
                                       
                                        
                                       
                                           
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     ξ 
                                   
                                   ) 
                                 
                               
                             
                             = 
                             
                               - 
                               
                                 2 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           n 
                                            
                                           
                                               
                                           
                                            
                                           ξ 
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
                  
                 
                     
                 
                  
                 
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         ′ 
                       
                     
                      
                     
                       L 
                       3 
                     
                      
                     
                       
                         J 
                         0 
                       
                        
                       
                         ( 
                         
                           
                              
                             k 
                              
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             = 
             
               8 
                
               π 
                
               
                   
               
                
               N 
                
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                     
                 
                  
                 
                   
                     
                       1 
                       
                         n 
                         2 
                       
                     
                      
                     
                       [ 
                       
                         1 
                         - 
                         
                           cos 
                            
                           
                             ( 
                             
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                    
                   
                     I 
                     
                       n 
                       , 
                       n 
                       , 
                       0 
                     
                   
                 
               
             
           
         
       
     
         [0207]    For the case ξ=π we get 
         [0000]    
       
         
           
             
               8 
                
               π 
                
               
                   
               
                
               N 
                
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                     
                 
                  
                 
                   
                     
                       1 
                       
                         n 
                         2 
                       
                     
                      
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           n 
                         
                       
                       ] 
                     
                   
                    
                   
                     I 
                     
                       0 
                       , 
                       n 
                       , 
                       n 
                     
                   
                 
               
             
             = 
             
               16 
                
               π 
                
               
                   
               
                
               N 
                
               
                 
                   ∑ 
                   
                     
                       n 
                       = 
                       1 
                     
                     , 
                     odd 
                   
                   ∞ 
                 
                  
                 
                     
                 
                  
                 
                   
                     1 
                     
                       n 
                       2 
                     
                   
                    
                   
                     I 
                     
                       n 
                       , 
                       n 
                       , 
                       0 
                     
                   
                 
               
             
           
         
       
     
         [0208]    We secondly look at the contribution of the second term of F: 
         [0000]    
       
         
           
             
               ∫ 
               0 
               ∞ 
             
              
             
               
                  
                 
                   z 
                   - 
                   r 
                 
                  
               
                
               
                   
               
                
               
                  
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
               
                
               
                 
                   ∫ 
                   0 
                   
                     2 
                      
                     π 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                      
                     
                       φ 
                       ′ 
                     
                   
                    
                   
                     L 
                     3 
                   
                    
                   
                     
                       ∑ 
                       
                         
                           n 
                           3 
                         
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       { 
                       
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               φ 
                             
                           
                            
                           
                             
                               J 
                               
                                 n 
                                 3 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                         + 
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               φ 
                             
                           
                            
                           
                             
                               J 
                               
                                 n 
                                 3 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     3 
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             
                               n 
                               3 
                             
                           
                            
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
           
         
       
       
         
           
             E 
             = 
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
                  
                 
                     
                 
                  
                 
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         ′ 
                       
                     
                      
                     N 
                     × 
                     
                       [ 
                       
                         
                           ∑ 
                           
                             
                               
                                 n 
                                 3 
                               
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             , 
                             
                               
                                 n 
                                 3 
                               
                               ≠ 
                               0 
                             
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               φ 
                             
                           
                            
                           
                             
                               J 
                               
                                 n 
                                 3 
                               
                             
                              
                             
                               ( 
                               
                                 
                                    
                                   k 
                                    
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 π 
                                 2 
                               
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 3 
                               
                                
                               
                                 φ 
                                 ′ 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                         
                       
                         
                           [ 
                           
                             
                               ∑ 
                               
                                 
                                   
                                     n 
                                     2 
                                   
                                   = 
                                   
                                     - 
                                     ∞ 
                                   
                                 
                                 , 
                                 
                                   
                                     n 
                                     2 
                                   
                                   ≠ 
                                   0 
                                 
                               
                               ∞ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                         
                                            
                                            
                                           
                                             ( 
                                             
                                               1 
                                               + 
                                               
                                                 n 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         π 
                                         2 
                                       
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             n 
                                             2 
                                           
                                            
                                           ξ 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 
                                   n 
                                   2 
                                 
                               
                                
                               
                                 
                                   J 
                                   
                                     n 
                                     2 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   
                                     φ 
                                     ′ 
                                   
                                 
                               
                             
                           
                           ] 
                         
                          
                         
                             
                           
                             
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       
                                         
                                           n 
                                           1 
                                         
                                         = 
                                         
                                           - 
                                           ∞ 
                                         
                                       
                                       , 
                                       
                                         
                                           n 
                                           1 
                                         
                                         ≠ 
                                         0 
                                       
                                     
                                     ∞ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         
                                            
                                           
                                             
                                                
                                                
                                               
                                                 ( 
                                                 
                                                   1 
                                                   + 
                                                   
                                                     n 
                                                     1 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               π 
                                               2 
                                             
                                           
                                         
                                          
                                         
                                           [ 
                                           
                                             1 
                                             - 
                                             
                                                
                                               
                                                  
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   n 
                                                   1 
                                                 
                                                  
                                                 ξ 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       
                                         n 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         J 
                                         
                                           n 
                                           1 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                              
                                             
                                               z 
                                               - 
                                               r 
                                             
                                              
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           n 
                                           1 
                                         
                                          
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                                
                               
                                 
 
                               
                                
                               E 
                             
                             = 
                             
                               
                                 ∫ 
                                 0 
                                 ∞ 
                               
                                
                               
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                  
                                 
                                   
                                     ∫ 
                                     0 
                                     
                                       2 
                                        
                                       π 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                        
                                       
                                         φ 
                                         ′ 
                                       
                                     
                                      
                                     N 
                                     × 
                                     
                                        
                                       
                                         
                                            
                                            
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                               - 
                                               
                                                 n 
                                                 2 
                                               
                                               - 
                                               
                                                 n 
                                                 3 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           π 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                       
                                         [ 
                                         
                                           1 
                                           - 
                                           
                                              
                                             
                                                
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 n 
                                                 1 
                                               
                                                
                                               ξ 
                                             
                                           
                                         
                                         ] 
                                       
                                       
                                         n 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         [ 
                                         
                                           1 
                                           - 
                                           
                                              
                                             
                                               
                                                 - 
                                                  
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 n 
                                                 2 
                                               
                                                
                                               ξ 
                                             
                                           
                                         
                                         ] 
                                       
                                       
                                         n 
                                         2 
                                       
                                     
                                      
                                     
                                        
                                       
                                          
                                          
                                         
                                             
                                         
                                          
                                         
                                           n 
                                           3 
                                         
                                          
                                         φ 
                                       
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               n 
                                               3 
                                             
                                             = 
                                             
                                               - 
                                               ∞ 
                                             
                                           
                                           , 
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               n 
                                               3 
                                             
                                             ≠ 
                                             0 
                                           
                                         
                                         ∞ 
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             J 
                                             
                                               n 
                                               1 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 k 
                                                 0 
                                               
                                                
                                               
                                                  
                                                 
                                                   z 
                                                   - 
                                                   r 
                                                 
                                                  
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           J 
                                           
                                             n 
                                             2 
                                           
                                         
                                          
                                         
                                             
                                           
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   0 
                                                 
                                                  
                                                 
                                                    
                                                   
                                                     z 
                                                     - 
                                                     r 
                                                   
                                                    
                                                 
                                               
                                               ) 
                                             
                                              
                                             
                                               
                                                 J 
                                                 
                                                   n 
                                                   3 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                      
                                                     k 
                                                      
                                                   
                                                    
                                                   
                                                      
                                                     
                                                       z 
                                                       - 
                                                       r 
                                                     
                                                      
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                                
                                               
                                                 
                                                   - 
                                                    
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   n 
                                                   1 
                                                 
                                                  
                                                 
                                                   φ 
                                                   ′ 
                                                 
                                               
                                             
                                              
                                             
                                                
                                               
                                                  
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   n 
                                                   2 
                                                 
                                                  
                                                 
                                                   φ 
                                                   ′ 
                                                 
                                               
                                             
                                              
                                             
                                                
                                               
                                                 
                                                   - 
                                                    
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   n 
                                                   3 
                                                 
                                                  
                                                 
                                                   φ 
                                                   ′ 
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0209]    We define and note: 
         [0000]    
       
         
           
             E 
             = 
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                    
                   
                     z 
                     - 
                     r 
                   
                    
                 
                  
                 
                    
                   
                      
                     
                       z 
                       - 
                       r 
                     
                      
                   
                 
                  
                 
                   
                     ∫ 
                     0 
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         ′ 
                       
                     
                      
                     N 
                     × 
                     
                        
                       
                         
                            
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                               - 
                               
                                 n 
                                 2 
                               
                               - 
                               
                                 n 
                                 3 
                               
                             
                             ) 
                           
                         
                          
                         
                           π 
                           2 
                         
                       
                     
                      
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                       
                         n 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                       
                         n 
                         2 
                       
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         
                           n 
                           3 
                         
                          
                         φ 
                       
                     
                      
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               
                                 n 
                                 3 
                               
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             , 
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               
                                 n 
                                 3 
                               
                               ≠ 
                               0 
                             
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             J 
                             
                               n 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 0 
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             J 
                             
                               n 
                               3 
                             
                           
                            
                           
                             ( 
                             
                               
                                  
                                 k 
                                  
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                          
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               3 
                             
                              
                             
                               φ 
                               ′ 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0210]    We now partition the summation into sections: 
         [0000]    
       
         
           
             E 
             = 
             
               2 
                
               π 
                
               
                   
               
                
               N 
               × 
               
                 { 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             
                               ∞ 
                               , 
                               ∞ 
                               , 
                               ∞ 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             
                               
                                 - 
                                 1 
                               
                               , 
                               ∞ 
                               , 
                               ∞ 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             
                               ∞ 
                               , 
                               
                                 - 
                                 1 
                               
                               , 
                               ∞ 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                             
                             
                               ∞ 
                               , 
                               ∞ 
                               , 
                               
                                 - 
                                 1 
                               
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             
                               
                                 - 
                                 1 
                               
                               , 
                               
                                 - 
                                 1 
                               
                               , 
                               ∞ 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                             
                             
                               ∞ 
                               , 
                               
                                 - 
                                 1 
                               
                               , 
                               
                                 - 
                                 1 
                               
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                             
                             
                               
                                 - 
                                 1 
                               
                               , 
                               ∞ 
                               , 
                               
                                 - 
                                 1 
                               
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               
                                 n 
                                 1 
                               
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             , 
                             
                               
                                 n 
                                 2 
                               
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             , 
                             
                               
                                 n 
                                 3 
                               
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                           
                           
                             
                               - 
                               1 
                             
                             , 
                             
                               - 
                               1 
                             
                             , 
                             
                               - 
                               1 
                             
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             M 
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 n 
                                 3 
                               
                             
                           
                            
                           
                             I 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 n 
                                 3 
                               
                             
                           
                            
                           
                             δ 
                             
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 + 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 n 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0211]    Changing the summations into ‘1 to ∞’ and changing the corresponding n index to −n: 
         [0000]    
       
         
           
             E 
             = 
             
               2 
                
               π 
                
               
                   
               
                
               N 
               × 
               
                 { 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   
                                     - 
                                     n 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   
                                     - 
                                     n 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               
                                 n 
                                 3 
                               
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           
                             M 
                             
                               
                                 
                                   - 
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                            
                           
                             I 
                             
                               
                                 - 
                                 
                                   n 
                                   1 
                                 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                            
                           
                             δ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0212]    Rearranging terms: 
         [0000]    
       
         
           
             E 
             = 
             
               2 
                
               π 
                
               
                   
               
                
               N 
               × 
               
                 { 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   
                                     - 
                                     n 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   
                                     - 
                                     n 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               
                                 
                                   
                                     - 
                                     n 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               
                                 n 
                                 3 
                               
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           
                             M 
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                            
                           
                             I 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                            
                           
                             δ 
                             
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0213]    Noting the properties of δ n,n′ : 
         [0000]    
       
         
           
             E 
             = 
             
               2 
                
               π 
                
               
                   
               
                
               N 
               × 
               
                 { 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               [ 
                               
                                 
                                   
                                     M 
                                     
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         n 
                                         3 
                                       
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         n 
                                         3 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     M 
                                     
                                       
                                         
                                           - 
                                           n 
                                         
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           3 
                                         
                                       
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         - 
                                         
                                           n 
                                           1 
                                         
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           3 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               [ 
                               
                                 
                                   
                                     M 
                                     
                                       
                                         
                                           - 
                                           n 
                                         
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         n 
                                         3 
                                       
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         - 
                                         
                                           n 
                                           1 
                                         
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         n 
                                         3 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     M 
                                     
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           3 
                                         
                                       
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           3 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                              
                             
                               δ 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   n 
                                   3 
                                 
                               
                             
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               
                                 n 
                                 3 
                               
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           
                             [ 
                             
                               
                                 
                                   M 
                                   
                                     
                                       
                                         - 
                                         n 
                                       
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                     , 
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     
                                       n 
                                       3 
                                     
                                   
                                 
                                  
                                 
                                   I 
                                   
                                     
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     , 
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     
                                       n 
                                       3 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   M 
                                   
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                     , 
                                     
                                       - 
                                       
                                         n 
                                         2 
                                       
                                     
                                     , 
                                     
                                       - 
                                       
                                         n 
                                         3 
                                       
                                     
                                   
                                 
                                  
                                 
                                   I 
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     
                                       - 
                                       
                                         n 
                                         2 
                                       
                                     
                                     , 
                                     
                                       - 
                                       
                                         n 
                                         3 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             δ 
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 + 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 n 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0214]    Using the δ: 
         [0000]    
       
         
           
             E 
             = 
             
               2 
                
               π 
                
               
                   
               
                
               N 
               × 
               
                 { 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   3 
                                 
                                 = 
                                 
                                   
                                     1 
                                     - 
                                     
                                       n 
                                       1 
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ≥ 
                                   1 
                                 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         M 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                        
                                       
                                         I 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       M 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 
                                   
                                     
                                       1 
                                        
                                       
                                         n 
                                         1 
                                       
                                     
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ≥ 
                                   1 
                                 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         M 
                                         
                                           
                                             - 
                                             
                                               n 
                                               1 
                                             
                                           
                                           , 
                                           
                                             - 
                                             
                                               n 
                                               2 
                                             
                                           
                                           , 
                                           
                                             
                                               n 
                                               1 
                                             
                                             - 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                        
                                       
                                         I 
                                         
                                           
                                             - 
                                             
                                               n 
                                               1 
                                             
                                           
                                           , 
                                           
                                             - 
                                             
                                               n 
                                               2 
                                             
                                           
                                           , 
                                           
                                             
                                               n 
                                               1 
                                             
                                             - 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       M 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             - 
                                             
                                               n 
                                               1 
                                             
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             - 
                                             
                                               n 
                                               1 
                                             
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               
                                 n 
                                 2 
                               
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       M 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     M 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0215]    Collecting terms for −n 1 +n 2 ≧1 and n 1 −n 2 ≧1 we get: 
         [0000]    
       
         
           
             E 
             = 
             
               2 
                
               π 
                
               
                   
               
                
               N 
               × 
               
                 { 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 
                                   
                                     1 
                                      
                                     
                                       n 
                                       2 
                                     
                                   
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                               
                               , 
                               
                                 ≠ 
                                 0 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         M 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                        
                                       
                                         I 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       M 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               
                                 n 
                                 2 
                               
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       M 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     M 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0216]    From the explicit expression of M n     1     ,n     2     ,n     3    we see that: 
         [0000]    
       
         
           
             E 
             = 
             
               2 
                
               π 
                
               
                   
               
                
               N 
               × 
               
                 { 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 
                                   
                                     1 
                                      
                                     
                                       n 
                                       2 
                                     
                                   
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                               
                               , 
                               
                                 ≠ 
                                 0 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         M 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                        
                                       
                                         I 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       M 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             - 
                                             
                                               n 
                                               1 
                                             
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                       * 
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               
                                 n 
                                 2 
                               
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       M 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             n 
                                             1 
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     M 
                                     
                                       
                                         - 
                                         
                                           n 
                                           1 
                                         
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         
                                           n 
                                           1 
                                         
                                         + 
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                     * 
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                       , 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0217]    From the properties I n     1     ,n     2     ,n     3   : 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 I 
                 
                   
                     - 
                     
                       n 
                       1 
                     
                   
                   , 
                   
                     - 
                     
                       n 
                       2 
                     
                   
                   , 
                   
                     
                       n 
                       1 
                     
                     - 
                     
                       n 
                       2 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       n 
                       1 
                     
                   
                    
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       n 
                       2 
                     
                   
                    
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       
                         - 
                         
                           n 
                           1 
                         
                       
                       + 
                       
                         n 
                         2 
                       
                     
                   
                    
                   
                     I 
                     
                       
                         n 
                         1 
                       
                       , 
                       
                         n 
                         2 
                       
                       , 
                       
                         
                           - 
                           
                             n 
                             1 
                           
                         
                         + 
                         
                           n 
                           2 
                         
                       
                     
                   
                 
                 = 
                 
                   I 
                   
                     
                       n 
                       1 
                     
                     , 
                     
                       
                         n 
                         2 
                       
                       - 
                       
                         n 
                         1 
                       
                       + 
                       
                         n 
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   I 
                   
                     
                       n 
                       1 
                     
                     , 
                     
                       - 
                       
                         n 
                         2 
                       
                     
                     , 
                     
                       
                         - 
                         
                           n 
                           1 
                         
                       
                       - 
                       
                         n 
                         2 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         n 
                         2 
                       
                     
                      
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         
                           - 
                           
                             n 
                             1 
                           
                         
                         + 
                         
                           n 
                           2 
                         
                       
                     
                      
                     
                       I 
                       
                         
                           n 
                           1 
                         
                         , 
                         
                           n 
                           2 
                         
                         , 
                         
                           
                             n 
                             1 
                           
                           + 
                           
                             n 
                             2 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         n 
                         1 
                       
                     
                      
                     
                       I 
                       
                         
                           n 
                           1 
                         
                         , 
                         
                           n 
                           2 
                         
                         , 
                         
                           
                             n 
                             1 
                           
                           + 
                           
                             n 
                             2 
                           
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 I 
                 
                   
                     - 
                     
                       n 
                       1 
                     
                   
                   , 
                   
                     n 
                     2 
                   
                   , 
                   
                     
                       n 
                       1 
                     
                     + 
                     
                       n 
                       2 
                     
                   
                 
               
               = 
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     n 
                     1 
                   
                 
                  
                 
                   I 
                   
                     
                       n 
                       1 
                     
                     , 
                     
                       n 
                       2 
                     
                     , 
                     
                       
                         n 
                         1 
                       
                       + 
                       
                         n 
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               E 
               = 
               
                 2 
                  
                 π 
                  
                 
                     
                 
                  
                 N 
                 × 
                 
                   { 
                   
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   
                                     n 
                                     2 
                                   
                                   = 
                                   
                                     
                                       1 
                                        
                                       
                                         n 
                                         2 
                                       
                                     
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                 
                                 , 
                                 
                                   ≠ 
                                   0 
                                 
                               
                               ∞ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 [ 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   Re 
                                    
                                   
                                       
                                   
                                    
                                   
                                     M 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         + 
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                                
                               
                                 I 
                                 
                                   
                                     n 
                                     1 
                                   
                                   , 
                                   
                                     n 
                                     2 
                                   
                                   , 
                                   
                                     
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 
                                   ( 
                                   
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 
                                   n 
                                   1 
                                 
                               
                                
                               
                                 [ 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   Re 
                                    
                                   
                                       
                                   
                                    
                                   
                                     M 
                                     
                                       
                                         - 
                                         
                                           n 
                                           1 
                                         
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         
                                           n 
                                           1 
                                         
                                         + 
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   
                                     n 
                                     1 
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   } 
                 
               
             
           
         
       
     
         [0218]    We now recall the expression for 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 M 
                 
                   
                     n 
                     1 
                   
                   , 
                   
                     n 
                     2 
                   
                   , 
                   
                     n 
                     3 
                   
                 
               
               = 
               
                 
                   
                      
                     
                       
                          
                          
                         
                           ( 
                           
                             
                               n 
                               1 
                             
                             - 
                             
                               n 
                               2 
                             
                             - 
                             
                               n 
                               3 
                             
                           
                           ) 
                         
                       
                        
                       
                         π 
                         2 
                       
                     
                   
                    
                   
                     [ 
                     
                       1 
                       - 
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             1 
                           
                            
                           ξ 
                         
                       
                       - 
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           ξ 
                         
                       
                       + 
                       
                          
                         
                           
                              
                              
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                            
                           ξ 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                    
                   
                      
                      
                     
                         
                     
                      
                     
                       n 
                       3 
                     
                      
                     φ 
                   
                 
               
             
           
         
       
       
         
           
             
               M 
               
                 
                   n 
                   1 
                 
                 , 
                 
                   n 
                   2 
                 
                 , 
                 
                   
                     - 
                     
                       n 
                       1 
                     
                   
                   + 
                   
                     n 
                     2 
                   
                 
               
             
             = 
             
               
                 
                   
                     
                        
                       
                         
                            
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                               - 
                               
                                 n 
                                 2 
                               
                               + 
                               
                                 n 
                                 1 
                               
                               - 
                               
                                 n 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           π 
                           2 
                         
                       
                     
                     
                       
                         n 
                         1 
                       
                        
                       
                         n 
                         2 
                       
                     
                   
                    
                   
                     [ 
                     
                       1 
                       - 
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             1 
                           
                            
                           ξ 
                         
                       
                       - 
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           ξ 
                         
                       
                       + 
                       
                          
                         
                           
                              
                              
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                            
                           ξ 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                    
                   
                     
                        
                        
                       
                         ( 
                         
                           
                             - 
                             
                               n 
                               1 
                             
                           
                           + 
                           
                             n 
                             2 
                           
                         
                         ) 
                       
                     
                      
                     φ 
                   
                 
                  
                 
                   
 
                 
                  
                 
                     
                 
                  
                 
                    
                   
                      
                      
                     
                         
                     
                      
                     n 
                      
                     
                         
                     
                      
                     π 
                   
                 
               
               = 
               
                 
                   
                     cos 
                      
                     
                       ( 
                       
                         n 
                          
                         
                             
                         
                          
                         π 
                       
                       ) 
                     
                   
                   + 
                   
                      
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           n 
                            
                           
                               
                           
                            
                           π 
                         
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     cos 
                      
                     
                       ( 
                       
                         n 
                          
                         
                             
                         
                          
                         π 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           n 
                         
                          
                         
                           
 
                         
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                           
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                           
                           + 
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       n 
                                       1 
                                     
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                       == 
                       
                         1 
                         - 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                         - 
                         
                            
                            
                           
                               
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                         + 
                         
                            
                            
                           
                               
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   n 
                                   2 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                         + 
                         
                            
                            
                           
                               
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       n 
                                       1 
                                     
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         - 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                         - 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                         + 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                         - 
                         
                           
                              
                              
                             
                               [ 
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         n 
                                         1 
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         n 
                                         2 
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             n 
                                             1 
                                           
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                         ) 
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
 
                           
                            
                           
                               
                           
                            
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               φ 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                                  
                                 φ 
                               
                               ) 
                             
                           
                           + 
                           
                              
                              
                             
                                 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         
                                           n 
                                           1 
                                         
                                       
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                    
                                   φ 
                                 
                                 ) 
                               
                             
                              
                             
                               
 
                             
                              
                             
                               ReM 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                             
                             
                               
                                 n 
                                 1 
                               
                                
                               
                                 n 
                                 2 
                               
                             
                           
                            
                           
                             { 
                             
                               
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             n 
                                             1 
                                           
                                            
                                           ξ 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             n 
                                             2 
                                           
                                            
                                           ξ 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                               - 
                                               
                                                 n 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                            
                                           ξ 
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             - 
                                             
                                               n 
                                               1 
                                             
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                         ) 
                                       
                                        
                                       φ 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                   
                                 
                                   
                                     [ 
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               2 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                 - 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                           ) 
                                         
                                          
                                         φ 
                                       
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0219]    We note that if 
         [0000]    
       
         
           
             
               ReM 
               
                 
                   n 
                   1 
                 
                 , 
                 
                   n 
                   2 
                 
                 , 
                 
                   
                     - 
                     
                       n 
                       1 
                     
                   
                   + 
                   
                     n 
                     2 
                   
                 
               
             
             = 
             
               
                 ReM 
                 
                   n 
                   , 
                   n 
                   , 
                   0 
                 
               
               = 
               
                 
                   1 
                   
                     n 
                     2 
                   
                 
                  
                 
                   { 
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             cos 
                              
                             
                               ( 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                           - 
                           
                             cos 
                              
                             
                               ( 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               ( 
                               0 
                               ) 
                             
                              
                             φ 
                           
                           ) 
                         
                       
                     
                     + 
                     
                         
                       
                         
                           
                             
                               [ 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   ξ 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             sin 
                              
                             
                               ( 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                           - 
                           
                             sin 
                              
                             
                                 
                             
                              
                             0 
                              
                             ξ 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             0 
                              
                             φ 
                           
                         
                         = 
                         
                           
                             
                               1 
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               221 
                             
                             - 
                             
                               cos 
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               ξ 
                                
                               
                                 
 
                               
                                
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               N 
                               × 
                               
                                 
                                   { 
                                   
                                     
                                       ∑ 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           
                                             n 
                                             2 
                                           
                                           = 
                                           1 
                                         
                                       
                                       ∞ 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         [ 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           Re 
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                               , 
                                               
                                                 n 
                                                 2 
                                               
                                               , 
                                               
                                                 
                                                   - 
                                                   
                                                     n 
                                                     1 
                                                   
                                                 
                                                 + 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                        
                                       
                                         I 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                   
                                   } 
                                 
                                 
                                   
                                     
                                       n 
                                       2 
                                     
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   , 
                                   
                                     = 
                                     0 
                                   
                                 
                               
                             
                           
                           = 
                           
                             8 
                              
                             π 
                              
                             
                                 
                             
                              
                             n 
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 ∞ 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     1 
                                     
                                       n 
                                       2 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             n 
                                              
                                             
                                                 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   I 
                                   
                                     n 
                                     , 
                                     n 
                                     , 
                                     0 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0220]    Which is exactly the result from the second isotropic term. Therefore, if we allow n 1 =n 2  we can ommit the second isotropic term. 
         [0221]    Similarly again 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 M 
                 
                   
                     n 
                     1 
                   
                   , 
                   
                     n 
                     2 
                   
                   , 
                   
                     n 
                     3 
                   
                 
               
               = 
               
                 
                   
                     
                       
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                         
                           
                             n 
                             1 
                           
                            
                           
                             n 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                           
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                           
                           + 
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       n 
                                       1 
                                     
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         
                           n 
                           3 
                         
                          
                         φ 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                   ⇒ 
                   
                     
 
                   
                    
                   
                     M 
                     
                       
                         - 
                         
                           n 
                           1 
                         
                       
                       , 
                       
                         n 
                         2 
                       
                       , 
                       
                         n 
                         3 
                       
                     
                   
                 
                 = 
                 
                   
                     - 
                     
                       
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                   - 
                                   
                                     n 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               π 
                               2 
                             
                           
                         
                         
                           
                             n 
                             1 
                           
                            
                           
                             n 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                           
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                           
                           + 
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       
                         n 
                         3 
                       
                        
                       φ 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               M 
               
                 
                   - 
                   
                     n 
                     1 
                   
                 
                 , 
                 
                   n 
                   2 
                 
                 , 
                 
                   
                     n 
                     1 
                   
                   + 
                   
                     n 
                     2 
                   
                 
               
             
             = 
             
               
                 
                   - 
                   
                     
                       
                          
                         
                           
                              
                              
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                            
                           π 
                         
                       
                       
                         
                           n 
                           1 
                         
                          
                         
                           n 
                           2 
                         
                       
                     
                      
                     
                       [ 
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             ξ 
                           
                         
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             ξ 
                           
                         
                         + 
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                              
                             ξ 
                           
                         
                       
                       ] 
                     
                   
                 
                  
                 
                   
                      
                     
                       
                          
                          
                         
                           ( 
                           
                             
                               n 
                               1 
                             
                             + 
                             
                               n 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       φ 
                     
                   
                    
                   
                     
 
                   
                   [ 
                   
                     M 
                     
                       
                         - 
                         
                           n 
                           1 
                         
                       
                       , 
                       
                         n 
                         2 
                       
                       , 
                       
                         
                           n 
                           1 
                         
                         + 
                         
                           n 
                           2 
                         
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   
                     - 
                     
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             
                               n 
                               1 
                             
                             + 
                             
                               n 
                               2 
                             
                           
                         
                         
                           
                             n 
                             1 
                           
                            
                           
                             n 
                             2 
                           
                         
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                           
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                           
                           + 
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               ξ 
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
                        
                       
                         
                            
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                               + 
                               
                                 n 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         φ 
                       
                     
                      
                     
                       
 
                     
                     [ 
                     
                       1 
                       - 
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             1 
                           
                            
                           ξ 
                         
                       
                       - 
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           ξ 
                         
                       
                       + 
                       
                          
                         
                           
                              
                              
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                            
                           ξ 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   
                     1 
                     - 
                     
                       cos 
                        
                       
                         ( 
                         
                           
                             - 
                             
                               n 
                               1 
                             
                           
                            
                           ξ 
                         
                         ) 
                       
                     
                     - 
                     
                        
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               - 
                               
                                 n 
                                 1 
                               
                             
                              
                             ξ 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       cos 
                        
                       
                         ( 
                         
                           
                             - 
                             
                               n 
                               2 
                             
                           
                            
                           ξ 
                         
                         ) 
                       
                     
                     - 
                     
                        
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               - 
                               
                                 n 
                                 2 
                               
                             
                              
                             ξ 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       cos 
                        
                       
                         ( 
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                 + 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                            
                           ξ 
                         
                         ) 
                       
                     
                     + 
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                 + 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                            
                           ξ 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       1 
                       - 
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               n 
                               1 
                             
                              
                             ξ 
                           
                           ) 
                         
                       
                       + 
                       
                          
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               n 
                               2 
                             
                              
                             ξ 
                           
                           ) 
                         
                       
                       + 
                       
                          
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                 + 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                              
                             ξ 
                           
                           ) 
                         
                       
                       - 
                       
                          
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     n 
                                     1 
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               ξ 
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                           - 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   n 
                                   2 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       n 
                                       1 
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       + 
                       
                          
                          
                         
                           [ 
                           
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                     1 
                                   
                                    
                                   ξ 
                                 
                                 ) 
                               
                             
                             + 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                     2 
                                   
                                    
                                   ξ 
                                 
                                 ) 
                               
                             
                             - 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         1 
                                       
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                    
                                   ξ 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                  
                 
                   
                      
                      
                     
                       ( 
                       
                         
                           n 
                           1 
                         
                         + 
                         
                           n 
                           2 
                         
                       
                       ) 
                     
                   
                    
                   φ 
                 
               
               = 
               
                 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               n 
                               1 
                             
                             + 
                             
                               n 
                               2 
                             
                           
                           ) 
                         
                          
                         φ 
                       
                       ) 
                     
                   
                   + 
                   
                      
                      
                     
                         
                     
                      
                     
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                 + 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                              
                             φ 
                           
                           ) 
                         
                       
                        
                       
                         
 
                       
                       [ 
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             ξ 
                           
                         
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             ξ 
                           
                         
                         + 
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                              
                             ξ 
                           
                         
                       
                       ] 
                     
                      
                     
                        
                       
                         
                            
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                               + 
                               
                                 n 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         φ 
                       
                     
                   
                 
                 = 
                 
                     
                   
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                           - 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   n 
                                   2 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       n 
                                       1 
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ) 
                                 
                                  
                                 ξ 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                 + 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                              
                             φ 
                           
                           ) 
                         
                       
                     
                     - 
                     
                         
                       
                         
                           
                             
                               [ 
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         n 
                                         1 
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         n 
                                         2 
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             n 
                                             1 
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                         ) 
                                       
                                        
                                       ξ 
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         1 
                                       
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                    
                                   φ 
                                 
                                 ) 
                               
                             
                              
                             
                               
 
                             
                              
                             
                               
                                  
                                  
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             n 
                                             1 
                                           
                                            
                                           ξ 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             n 
                                             2 
                                           
                                            
                                           ξ 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                               + 
                                               
                                                 n 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                            
                                           ξ 
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           n 
                                           1 
                                         
                                         + 
                                         
                                           n 
                                           2 
                                         
                                       
                                       ) 
                                     
                                      
                                     φ 
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                             
                                
                                
                               
                                 [ 
                                 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           n 
                                           1 
                                         
                                          
                                         ξ 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           n 
                                           2 
                                         
                                          
                                         ξ 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                           ) 
                                         
                                          
                                         ξ 
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         1 
                                       
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                    
                                   φ 
                                 
                                 ) 
                               
                             
                              
                             
                               
 
                             
                              
                             
                               Re 
                                
                               
                                 [ 
                                 
                                   M 
                                   
                                     
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     , 
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     
                                       
                                         n 
                                         1 
                                       
                                       + 
                                       
                                         n 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         = 
                         
                           
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 
                                   
                                     n 
                                     1 
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                               
                             
                           
                            
                           
                             { 
                             
                               
                                 
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               2 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                 + 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                           ) 
                                         
                                          
                                         φ 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                       
                                     
                                       
                                         [ 
                                         
                                           
                                             - 
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     n 
                                                     1 
                                                   
                                                    
                                                   ξ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           - 
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   2 
                                                 
                                                  
                                                 ξ 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       n 
                                                       1 
                                                     
                                                     + 
                                                     
                                                       n 
                                                       2 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 ξ 
                                               
                                               ) 
                                             
                                           
                                         
                                         ] 
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                 + 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                              
                                             φ 
                                           
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                    
                                   
                                     
 
                                   
                                    
                                   E 
                                 
                               
                               = 
                               
                                 
                                   4 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   N 
                                    
                                   
                                     
                                       ∑ 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           
                                             n 
                                             2 
                                           
                                           = 
                                           1 
                                         
                                       
                                       ∞ 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                           
                                             
                                               n 
                                               1 
                                             
                                             - 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                         
                                           
                                             n 
                                             1 
                                           
                                            
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             
                                               [ 
                                               
                                                 1 
                                                 - 
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         1 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 - 
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         2 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 + 
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         ( 
                                                         
                                                           
                                                             n 
                                                             1 
                                                           
                                                           - 
                                                           
                                                             n 
                                                             2 
                                                           
                                                         
                                                         ) 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                              
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         - 
                                                         
                                                           n 
                                                           1 
                                                         
                                                       
                                                       + 
                                                       
                                                         n 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   φ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                           
                                             
                                               [ 
                                               
                                                 
                                                   sin 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         1 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 - 
                                                 
                                                   sin 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         2 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 - 
                                                 
                                                   sin 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         ( 
                                                         
                                                           
                                                             n 
                                                             1 
                                                           
                                                           - 
                                                           
                                                             n 
                                                             2 
                                                           
                                                         
                                                         ) 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                              
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         - 
                                                         
                                                           n 
                                                           1 
                                                         
                                                       
                                                       + 
                                                       
                                                         n 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   φ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         } 
                                       
                                        
                                       
                                         I 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   4 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   N 
                                    
                                   
                                     
                                       ∑ 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           
                                             n 
                                             2 
                                           
                                           = 
                                           1 
                                         
                                       
                                       ∞ 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                           
                                             n 
                                             2 
                                           
                                         
                                         
                                           
                                             n 
                                             1 
                                           
                                            
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             
                                               [ 
                                               
                                                 1 
                                                 - 
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         1 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 - 
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         2 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 + 
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         ( 
                                                         
                                                           
                                                             n 
                                                             1 
                                                           
                                                           + 
                                                           
                                                             n 
                                                             2 
                                                           
                                                         
                                                         ) 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                              
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         1 
                                                       
                                                       + 
                                                       
                                                         n 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   φ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                           
                                             
                                               [ 
                                               
                                                 
                                                   - 
                                                   
                                                     sin 
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           n 
                                                           1 
                                                         
                                                          
                                                         ξ 
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 - 
                                                 
                                                   sin 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         2 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 + 
                                                 
                                                   sin 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         ( 
                                                         
                                                           
                                                             n 
                                                             1 
                                                           
                                                           + 
                                                           
                                                             n 
                                                             2 
                                                           
                                                         
                                                         ) 
                                                       
                                                        
                                                       ξ 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                              
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         1 
                                                       
                                                       + 
                                                       
                                                         n 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   φ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         } 
                                       
                                        
                                       
                                         I 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             
                                               n 
                                               1 
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0222]    For the case ξ=π we get 
         [0000]    
       
         
           
             
               [ 
               
                 1 
                 - 
                 
                   cos 
                    
                   
                     ( 
                     
                       
                         n 
                         1 
                       
                        
                       π 
                     
                     ) 
                   
                 
                 - 
                 
                   cos 
                    
                   
                     ( 
                     
                       
                         n 
                         2 
                       
                        
                       π 
                     
                     ) 
                   
                 
                 + 
                 
                   cos 
                    
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             n 
                             1 
                           
                           - 
                           
                             n 
                             2 
                           
                         
                         ) 
                       
                        
                       π 
                     
                     ) 
                   
                 
               
               ] 
             
             = 
             
                 
               
                 
                   [ 
                   
                     1 
                     - 
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         n 
                         1 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         n 
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         
                           n 
                           1 
                         
                         - 
                         
                           n 
                           2 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     4 
                                     , 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         
                                           n 
                                           2 
                                         
                                          
                                         
                                             
                                         
                                          
                                         both 
                                          
                                         
                                             
                                         
                                          
                                         odd 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 
                                   
                                     0 
                                     , 
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                               , 
                                               
                                                 
                                                   n 
                                                   2 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 both 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 even 
                                               
                                             
                                             ) 
                                           
                                            
                                           or 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               odd 
                                                
                                               
                                                   
                                               
                                                
                                               and 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 n 
                                                 2 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               even 
                                             
                                             ) 
                                           
                                            
                                           or 
                                         
                                       
                                     
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             even 
                                              
                                             
                                                 
                                             
                                              
                                             and 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               n 
                                               2 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             odd 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                              
                             
                               
 
                             
                              
                             
                                 
                             
                             [ 
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       n 
                                       1 
                                     
                                      
                                     π 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         n 
                                         2 
                                       
                                        
                                       π 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             n 
                                             1 
                                           
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                         ) 
                                       
                                        
                                       π 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           = 
                           0 
                         
                         ; 
                       
                        
                       
                         
 
                       
                        
                       
                           
                       
                       [ 
                       
                         
                           - 
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 π 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                                
                               π 
                             
                             ) 
                           
                         
                         + 
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     n 
                                     1 
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               π 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                     = 
                     0 
                   
                 
               
             
           
         
       
     
         [0223]    We note that for odd n 1 ,n 2  we get (−1) n     2   =−1 and (−1) n     1     −n     2   =1, therefore: 
         [0000]    
       
         
           
             E 
             = 
             
               16 
                
               π 
                
               
                   
               
                
               
                 N 
                 [ 
                 
                   
                     
                       ∑ 
                       
                         
                           n 
                           1 
                         
                         , 
                         
                           
                             n 
                             2 
                           
                           = 
                           1 
                         
                         , 
                         odd 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         1 
                         
                           
                             n 
                             1 
                           
                            
                           
                             n 
                             2 
                           
                         
                       
                        
                       
                           
                         
                           [ 
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         + 
                                         
                                           n 
                                           2 
                                         
                                       
                                       ) 
                                     
                                      
                                     φ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     n 
                                     1 
                                   
                                   , 
                                   
                                     n 
                                     2 
                                   
                                   , 
                                   
                                     
                                       - 
                                       
                                         n 
                                         1 
                                       
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           n 
                                           1 
                                         
                                         + 
                                         
                                           n 
                                           2 
                                         
                                       
                                       ) 
                                     
                                      
                                     φ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     n 
                                     1 
                                   
                                   , 
                                   
                                     n 
                                     2 
                                   
                                   , 
                                   
                                     
                                       n 
                                       1 
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ] 
                       
                        
                       
                         
 
                       
                        
                       
                         
                           H 
                           BF 
                         
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         
                           
                              
                             2 
                           
                            
                           
                             ( 
                             
                               z 
                               - 
                               r 
                             
                             ) 
                           
                         
                          
                         
                           
                             h 
                             BF 
                           
                            
                           
                             ( 
                             
                               z 
                               - 
                               r 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               · 
                               
                                 [ 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         N 
                          
                         
                             
                         
                          
                         
                           ξ 
                           2 
                         
                          
                         
                           
                             I 
                             
                               0 
                               , 
                               0 
                               , 
                               0 
                             
                           
                           ++ 
                         
                          
                         8 
                          
                         π 
                          
                         
                             
                         
                          
                         N 
                          
                         
                             
                         
                          
                         ξ 
                          
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 1 
                               
                               , 
                               even 
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 1 
                                 n 
                               
                                
                               
                                 [ 
                                 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         φ 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           n 
                                            
                                           
                                               
                                           
                                            
                                           φ 
                                         
                                         - 
                                         
                                           n 
                                            
                                           
                                               
                                           
                                            
                                           ξ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               I 
                               
                                 0 
                                 , 
                                 n 
                                 , 
                                 n 
                               
                             
                           
                         
                       
                       + 
                       
                         4 
                          
                         π 
                          
                         
                             
                         
                          
                         N 
                          
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 
                                   ( 
                                   
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                               
                             
                              
                             
                               { 
                               
                                 
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               2 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                 - 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                    
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             - 
                                             
                                               n 
                                               1 
                                             
                                           
                                           + 
                                           
                                             n 
                                             2 
                                           
                                         
                                         ) 
                                       
                                        
                                       φ 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     [ 
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               2 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                 - 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                           ) 
                                         
                                          
                                         φ 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               } 
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       - 
                       
                         4 
                          
                         π 
                          
                         
                             
                         
                          
                         N 
                          
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 1 
                               
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 
                                   ( 
                                   
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 
                                   n 
                                   2 
                                 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                               
                             
                              
                             
                               { 
                               
                                 
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               2 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                 + 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                           ) 
                                         
                                          
                                         φ 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     [ 
                                     
                                       
                                         - 
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                                
                                               ξ 
                                             
                                             ) 
                                           
                                         
                                       
                                       - 
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               n 
                                               2 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                 + 
                                                 
                                                   n 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                              
                                             ξ 
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               n 
                                               1 
                                             
                                             + 
                                             
                                               n 
                                               2 
                                             
                                           
                                           ) 
                                         
                                          
                                         φ 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               } 
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   
                                     n 
                                     1 
                                   
                                   + 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0224]    For the case 
         [0000]    
       
         
           
             
               
                 
                   H 
                   BF 
                 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
                
               
                 | 
                 
                   ξ 
                   = 
                   π 
                 
               
             
             = 
             
               
                 2 
                  
                 
                   π 
                   3 
                 
                  
                 
                   NI 
                   
                     0 
                     , 
                     0 
                     , 
                     0 
                   
                 
               
               + 
               
                 16 
                  
                 π 
                  
                 
                     
                 
                  
                 
                   N 
                   [ 
                   
                     
                       ∑ 
                       
                         
                           
                             n 
                             1 
                           
                           , 
                           
                             
                               n 
                               2 
                             
                             = 
                             1 
                           
                         
                         odd 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         1 
                         
                           
                             n 
                             1 
                           
                            
                           
                             n 
                             2 
                           
                         
                       
                        
                       
                           
                         
                           [ 
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           n 
                                           2 
                                         
                                         - 
                                         
                                           n 
                                           1 
                                         
                                       
                                       ) 
                                     
                                      
                                     φ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     n 
                                     1 
                                   
                                   , 
                                   
                                     n 
                                     2 
                                   
                                   , 
                                   
                                     
                                       - 
                                       
                                         n 
                                         2 
                                       
                                     
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           n 
                                           1 
                                         
                                         + 
                                         
                                           n 
                                           2 
                                         
                                       
                                       ) 
                                     
                                      
                                     φ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     n 
                                     1 
                                   
                                   , 
                                   
                                     n 
                                     2 
                                   
                                   , 
                                   
                                     
                                       n 
                                       1 
                                     
                                     + 
                                     
                                       n 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
       ANNEX A III 
     Integration to obtain H BF (k) for ξ=π 
       [0225]    We repeat the calculation for ξ=π by starting with the more general form as a check: 
         [0000]    
       
         
           
             
               
                 H 
                 BF 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               N 
                
               
                 
                   ∑ 
                   
                     
                       n 
                       2 
                     
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   ∞ 
                 
                  
                 
                     
                 
                  
                 
                   
                     ∑ 
                     
                       
                         n 
                         1 
                       
                       = 
                       
                         - 
                         ∞ 
                       
                     
                     ∞ 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         2 
                          
                         
                           π 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                 1 
                               
                             
                             - 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           n 
                           1 
                         
                          
                         
                           n 
                           2 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                     1 
                                   
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                              
                             π 
                           
                         
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                              
                             π 
                           
                         
                         - 
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               n 
                               1 
                             
                              
                             π 
                           
                         
                         + 
                         1 
                       
                       ) 
                     
                      
                     
                        
                       
                         
                            
                            
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                               - 
                               
                                 n 
                                 1 
                               
                             
                             ) 
                           
                         
                          
                         φ 
                       
                     
                      
                     
                       I 
                       
                         
                           n 
                           1 
                         
                         , 
                         
                           n 
                           2 
                         
                         , 
                         
                           
                             n 
                             2 
                           
                           - 
                           
                             n 
                             1 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0226]    Note the identity: 
         [0000]    
       
         
           
             
                
               
                  
                  
                 
                     
                 
                  
                 n 
                  
                 
                     
                 
                  
                 π 
               
             
             = 
             
               
                 
                   cos 
                    
                   
                     ( 
                     
                       n 
                        
                       
                           
                       
                        
                       π 
                     
                     ) 
                   
                 
                 + 
                 
                    
                    
                   
                       
                   
                    
                   
                     sin 
                      
                     
                       ( 
                       
                         n 
                          
                         
                             
                         
                          
                         π 
                       
                       ) 
                     
                   
                 
               
               = 
               
                 { 
                 
                   
                     
                       
                         1 
                         , 
                       
                     
                     
                       
                         n 
                         = 
                         even 
                       
                     
                   
                   
                     
                       
                         
                           - 
                           1 
                         
                         , 
                       
                     
                     
                       
                         n 
                         = 
                         odd 
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    and use it for inspecting the n 1 =0 and n 2 =0 cases: 
         [0000]    
       
         
           
             
               
                 
                   2 
                    
                   
                     π 
                     
                       
                         n 
                         1 
                       
                       - 
                       
                         2 
                          
                         
                           n 
                           2 
                         
                       
                     
                   
                 
                 
                   
                     n 
                     1 
                   
                    
                   
                     n 
                     2 
                   
                 
               
                
               
                 ( 
                 
                   
                      
                     
                       
                          
                          
                         
                           ( 
                           
                             
                               n 
                               1 
                             
                             - 
                             
                               n 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       π 
                     
                   
                   - 
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         n 
                         2 
                       
                        
                       π 
                     
                   
                   - 
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       
                         n 
                         1 
                       
                        
                       π 
                     
                   
                   + 
                   1 
                 
                 ) 
               
             
             == 
             
               { 
               
                 
                   
                     
                       2 
                        
                       
                         π 
                         3 
                       
                     
                   
                   
                     
                       
                         
                           n 
                           1 
                         
                         = 
                         0 
                       
                       , 
                       
                         
                           n 
                           2 
                         
                         = 
                         0 
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                            
                           
                             
                               - 
                               2 
                             
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                           
                         
                          
                         
                           
                             
                               π 
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                                  
                                 π 
                               
                             
                             - 
                             π 
                           
                           
                             n 
                             2 
                           
                         
                       
                       ; 
                     
                   
                   
                     
                       { 
                       
                         
                           
                             
                               0 
                               ; 
                             
                           
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 0 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     even 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   
                                     π 
                                     2 
                                   
                                 
                                 
                                   n 
                                   2 
                                 
                               
                               ; 
                             
                           
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 0 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     odd 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                            
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               n 
                               2 
                             
                           
                         
                          
                         
                           
                             
                               - 
                               
                                 π 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     1 
                                   
                                    
                                   π 
                                 
                               
                             
                             + 
                             π 
                           
                           
                             n 
                             1 
                           
                         
                       
                       ; 
                     
                   
                   
                     
                       { 
                       
                         
                           
                             
                               0 
                               ; 
                             
                           
                           
                             
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 0 
                               
                               , 
                               
                                 
                                   n 
                                   1 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     even 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     
                                       π 
                                       2 
                                     
                                   
                                   
                                     n 
                                     1 
                                   
                                 
                               
                               ; 
                             
                           
                           
                             
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 0 
                               
                               , 
                               
                                 
                                   n 
                                   1 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     odd 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           2 
                            
                           
                             π 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   1 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             n 
                             1 
                           
                            
                           
                             n 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                    
                                   
                                     
                                        
                                        
                                       
                                         ( 
                                         
                                           
                                             n 
                                             1 
                                           
                                           - 
                                           
                                             n 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     π 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                      
                                     π 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       1 
                                     
                                      
                                     π 
                                   
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       { 
                       
                         
                           
                             
                               0 
                               ; 
                             
                           
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     even 
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     even 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               ; 
                             
                           
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     odd 
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     even 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               ; 
                             
                           
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     even 
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     odd 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 8 
                                  
                                 π 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     odd 
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 ≠ 
                                 
                                   0 
                                    
                                   
                                     ( 
                                     odd 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0227]    Therefore: 
         [0000]    
       
         
           
             
               
                 H 
                 BF 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               N 
                
               
                 { 
                 
                   
                     2 
                      
                     
                       π 
                       3 
                     
                      
                     
                       I 
                       
                         0 
                         , 
                         0 
                         , 
                         0 
                       
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         
                           - 
                           
                             ∞ 
                              
                             
                               ( 
                               
                                 odd 
                                 , 
                                 
                                   
                                     n 
                                     2 
                                   
                                   ≠ 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           4 
                            
                           
                               
                           
                            
                           
                             π 
                             2 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           
                             n 
                             2 
                           
                            
                           φ 
                         
                       
                        
                       
                         I 
                         
                           0 
                           , 
                           
                             n 
                             2 
                           
                           , 
                           
                             n 
                             2 
                           
                         
                       
                     
                   
                   - 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         
                           - 
                           
                             ∞ 
                              
                             
                               ( 
                               
                                 odd 
                                 , 
                                 
                                   
                                     n 
                                     1 
                                   
                                   ≠ 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           4 
                            
                           
                               
                           
                            
                           
                             π 
                             2 
                           
                         
                         
                           n 
                           1 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             n 
                             1 
                           
                            
                           φ 
                         
                       
                        
                       
                         I 
                         
                           
                             n 
                             1 
                           
                           , 
                           0 
                           , 
                           
                             - 
                             
                               n 
                               1 
                             
                           
                         
                       
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         
                           - 
                           
                             ∞ 
                              
                             
                               ( 
                               
                                 odd 
                                 , 
                                 
                                   
                                     n 
                                     2 
                                   
                                   ≠ 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         ∑ 
                         
                           
                             n 
                             1 
                           
                           = 
                           
                             - 
                             
                               ∞ 
                                
                               
                                 ( 
                                 
                                   odd 
                                   , 
                                   
                                     
                                       n 
                                       1 
                                     
                                     ≠ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             8 
                              
                             π 
                           
                           
                             
                               n 
                               1 
                             
                              
                             
                               n 
                               2 
                             
                           
                         
                          
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                     2 
                                   
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                              
                             φ 
                           
                         
                          
                         
                           I 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               
                                 n 
                                 2 
                               
                               - 
                               
                                 n 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   I 
                   
                     
                       n 
                       1 
                     
                     , 
                     0 
                     , 
                     
                       - 
                       
                         n 
                         1 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       n 
                       1 
                     
                   
                    
                   
                     I 
                     
                       
                         n 
                         1 
                       
                       , 
                       0 
                       , 
                       
                         n 
                         1 
                       
                     
                   
                 
               
               ; 
               
                 
                   I 
                   
                     
                       n 
                       1 
                     
                     , 
                     0 
                     , 
                     
                       n 
                       1 
                     
                   
                 
                 = 
                 
                   I 
                   
                     0 
                     , 
                     
                       n 
                       1 
                     
                     , 
                     , 
                     
                       n 
                       1 
                     
                   
                 
               
             
           
         
       
     
         [0228]    The “second term” T 2  becomes: 
         [0000]    
       
         
           
             
               T 
               2 
             
             = 
             
               
                 
                   
                     ∑ 
                     
                       
                         n 
                         2 
                       
                       = 
                       
                         - 
                         
                           ∞ 
                            
                           
                             ( 
                             
                               odd 
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 ≠ 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                     
                     ∞ 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         4 
                          
                         
                             
                         
                          
                         
                           π 
                           2 
                         
                       
                       
                         n 
                         2 
                       
                     
                      
                     
                        
                       
                          
                          
                         
                             
                         
                          
                         
                           n 
                           2 
                         
                          
                         φ 
                       
                     
                      
                     
                       I 
                       
                         0 
                         , 
                         
                           n 
                           2 
                         
                         , 
                         
                           n 
                           2 
                         
                       
                     
                   
                 
                 - 
                 
                   
                     ∑ 
                     
                       
                         n 
                         1 
                       
                       = 
                       
                         - 
                         
                           ∞ 
                            
                           
                             ( 
                             
                               odd 
                               , 
                               
                                 
                                   n 
                                   1 
                                 
                                 ≠ 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                     
                     ∞ 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         4 
                          
                         
                             
                         
                          
                         
                           π 
                           2 
                         
                       
                       
                         n 
                         1 
                       
                     
                      
                     
                        
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         
                           n 
                           1 
                         
                          
                         φ 
                       
                     
                      
                     
                       I 
                       
                         
                           n 
                           1 
                         
                         , 
                         0 
                         , 
                         
                           - 
                           
                             n 
                             1 
                           
                         
                       
                     
                   
                 
               
               = 
               
                 
                   
                     ∑ 
                     
                       
                         n 
                         2 
                       
                       = 
                       
                         - 
                         
                           ∞ 
                            
                           
                             ( 
                             
                               odd 
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 ≠ 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                     
                     ∞ 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         4 
                          
                         
                             
                         
                          
                         
                           π 
                           2 
                         
                       
                       
                         n 
                         2 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               φ 
                             
                           
                            
                           
                             I 
                             
                               0 
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 n 
                                 2 
                               
                             
                           
                         
                         - 
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 n 
                                 2 
                               
                                
                               φ 
                             
                           
                            
                           
                             I 
                             
                               
                                 n 
                                 2 
                               
                               , 
                               0 
                               , 
                               
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         
                           n 
                           2 
                         
                         = 
                         
                           1 
                            
                           
                             ( 
                             odd 
                             ) 
                           
                         
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           4 
                            
                           
                               
                           
                            
                           
                             π 
                             2 
                           
                         
                         
                           n 
                           2 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                                  
                                 φ 
                               
                             
                              
                             
                               I 
                               
                                 0 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                               
                             
                           
                           - 
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                                  
                                 φ 
                               
                             
                              
                             
                               I 
                               
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 0 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                             
                           
                           - 
                           
                             
                                
                               
                                 
                                   - 
                                   
                                     
                                        
                                        
                                       
                                           
                                       
                                     
                                     2 
                                   
                                 
                                  
                                 φ 
                               
                             
                              
                             
                               I 
                               
                                 0 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                                  
                                 φ 
                               
                             
                              
                             
                               I 
                               
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 , 
                                 0 
                                 , 
                                 
                                   n 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             2 
                           
                           = 
                           
                             1 
                              
                             
                               ( 
                               odd 
                               ) 
                             
                           
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             4 
                              
                             
                                 
                             
                              
                             
                               π 
                               2 
                             
                           
                           
                             n 
                             2 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   φ 
                                 
                               
                                
                               
                                 I 
                                 
                                   0 
                                   , 
                                   
                                     n 
                                     2 
                                   
                                   , 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                             
                             - 
                             
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   φ 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     n 
                                     2 
                                   
                                   , 
                                   0 
                                   , 
                                   
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                 
                               
                             
                             - 
                             
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       n 
                                        
                                       
                                           
                                       
                                     
                                     2 
                                   
                                    
                                   φ 
                                 
                               
                                
                               
                                 I 
                                 
                                   0 
                                   , 
                                   
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   , 
                                   
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                  
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                    
                                   φ 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     - 
                                     
                                       n 
                                       2 
                                     
                                   
                                   , 
                                   0 
                                   , 
                                   
                                     n 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               2 
                             
                             = 
                             
                               1 
                                
                               
                                 ( 
                                 odd 
                                 ) 
                               
                             
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               4 
                                
                               
                                   
                               
                                
                               
                                 π 
                                 2 
                               
                             
                             
                               n 
                               2 
                             
                           
                            
                           
                             
                               I 
                               
                                 0 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                      
                                     φ 
                                   
                                 
                                 + 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                      
                                     φ 
                                   
                                 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                      
                                     φ 
                                   
                                 
                                 - 
                                 
                                    
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                      
                                     φ 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                       = 
                       
                         
                           0 
                            
                           
                             
 
                           
                            
                           
                               
                           
                           ⇒ 
                           
                             
 
                           
                            
                           
                             
                               H 
                               BF 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         = 
                         
                           N 
                            
                           
                             { 
                             
                               
                                 2 
                                  
                                 
                                   π 
                                   3 
                                 
                                  
                                 
                                   I 
                                   
                                     0 
                                     , 
                                     0 
                                     , 
                                     0 
                                   
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     
                                       n 
                                       2 
                                     
                                     = 
                                     
                                       - 
                                       
                                         ∞ 
                                          
                                         
                                           ( 
                                           
                                             odd 
                                             , 
                                             
                                               
                                                 n 
                                                 2 
                                               
                                               ≠ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ∞ 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       
                                         n 
                                         1 
                                       
                                       = 
                                       
                                         - 
                                         
                                           ∞ 
                                            
                                           
                                             ( 
                                             
                                               odd 
                                               , 
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                 ≠ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ∞ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         8 
                                          
                                         π 
                                       
                                       
                                         
                                           n 
                                           1 
                                         
                                          
                                         
                                           n 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                        
                                       
                                         
                                            
                                            
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 2 
                                               
                                               - 
                                               
                                                 n 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         φ 
                                       
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           
                                             n 
                                             2 
                                           
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0229]    For the double sum term we note that for the case: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   n 
                   2 
                 
                 = 
                 
                   
                     
                       - 
                       
                         n 
                         2 
                       
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       n 
                       1 
                     
                   
                   = 
                   
                     - 
                     
                       n 
                       1 
                     
                   
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               
                 
                   n 
                   1 
                 
                  
                 
                   ( 
                   odd 
                   ) 
                 
               
               , 
               
                 
                   
                     n 
                     2 
                   
                    
                   
                     ( 
                     odd 
                     ) 
                   
                 
                  
                 
                   
                     
                       : 
                     
                      
                     
                         
                     
                     [ 
                     
                       
                         
                           J 
                           
                             - 
                             n 
                           
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           n 
                         
                          
                         
                           
                             J 
                             n 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   14 
                 
               
             
           
         
       
       
         
           
             
               I 
               
                 
                   - 
                   
                     n 
                     1 
                   
                 
                 , 
                 
                   - 
                   
                     n 
                     2 
                   
                 
                 , 
                 
                   
                     - 
                     
                       n 
                       2 
                     
                   
                   + 
                   
                     n 
                     1 
                   
                 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     
                       n 
                       1 
                     
                     + 
                     
                       n 
                       2 
                     
                     + 
                     
                       n 
                       2 
                     
                     - 
                     
                       n 
                       1 
                     
                   
                 
                  
                 
                   I 
                   
                     
                       n 
                       1 
                     
                     , 
                     
                       n 
                       2 
                     
                     , 
                     
                       
                         n 
                         2 
                       
                       - 
                       
                         n 
                         1 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       2 
                        
                       
                           
                       
                        
                       
                         n 
                         2 
                       
                     
                   
                    
                   
                     I 
                     
                       
                         n 
                         1 
                       
                       , 
                       
                         n 
                         2 
                       
                       , 
                       
                         
                           n 
                           2 
                         
                         - 
                         
                           n 
                           1 
                         
                       
                     
                   
                 
                 = 
                 
                   
                     I 
                     
                       
                         n 
                         1 
                       
                       , 
                       
                         n 
                         2 
                       
                       , 
                       
                         
                           n 
                           2 
                         
                         - 
                         
                           n 
                           1 
                         
                       
                     
                   
                   = 
                   
                     
                       &gt; 
                       
                         
                           
                             
                               8 
                                
                               π 
                             
                             
                               
                                 n 
                                 1 
                               
                                
                               
                                 n 
                                 2 
                               
                             
                           
                            
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       n 
                                       2 
                                     
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               φ 
                             
                           
                            
                           
                             I 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 - 
                                 
                                   n 
                                   1 
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               8 
                                
                               π 
                             
                             
                               
                                 ( 
                                 
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       n 
                                       2 
                                     
                                     - 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               φ 
                             
                           
                            
                           
                             I 
                             
                               
                                 - 
                                 
                                   n 
                                   1 
                                 
                               
                               , 
                               
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               , 
                               
                                 
                                   - 
                                   
                                     n 
                                     2 
                                   
                                 
                                 + 
                                 
                                   n 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             8 
                              
                             π 
                           
                           
                             
                               n 
                               1 
                             
                              
                             
                               n 
                               2 
                             
                           
                         
                          
                         
                           
                             I 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 - 
                                 
                                   n 
                                   1 
                                 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                  
                                 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         
                                           n 
                                           2 
                                         
                                         - 
                                         
                                           n 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   φ 
                                 
                               
                               + 
                               
                                  
                                 
                                   
                                     - 
                                     
                                        
                                        
                                       
                                         ( 
                                         
                                           
                                             n 
                                             2 
                                           
                                           - 
                                           
                                             n 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   φ 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             16 
                              
                             π 
                           
                           
                             
                               n 
                               1 
                             
                              
                             
                               n 
                               2 
                             
                           
                         
                          
                         
                           I 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               
                                 n 
                                 2 
                               
                               - 
                               
                                 n 
                                 1 
                               
                             
                           
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     n 
                                     2 
                                   
                                   - 
                                   
                                     n 
                                     1 
                                   
                                 
                                 ) 
                               
                                
                               φ 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]      14  http://en.wikipedia.org/wiki/Bessel_function
 
and we note that for the case:
 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   ( 
                   
                     
                       n 
                       2 
                     
                     = 
                     
                       
                         
                           n 
                           2 
                         
                          
                         
                             
                         
                          
                         and 
                          
                         
                             
                         
                          
                         
                           n 
                           1 
                         
                       
                       = 
                       
                         - 
                         
                           n 
                           1 
                         
                       
                     
                   
                   ) 
                 
                 + 
                 
                   ( 
                   
                     
                       n 
                       2 
                     
                     = 
                     
                       
                         
                           - 
                           
                             n 
                             2 
                           
                         
                          
                         
                             
                         
                          
                         and 
                          
                         
                             
                         
                          
                         
                           n 
                           1 
                         
                       
                       = 
                       
                         n 
                         1 
                       
                     
                   
                   ) 
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               
                 
                   n 
                   1 
                 
                  
                 
                   ( 
                   odd 
                   ) 
                 
               
               , 
               
                 
                   
                     n 
                     2 
                   
                    
                   
                     ( 
                     odd 
                     ) 
                   
                 
                  
                 
                   : 
                 
               
             
           
         
       
       
         
           
             
               I 
               
                 
                   - 
                   
                     n 
                     1 
                   
                 
                 , 
                 
                   n 
                   2 
                 
                 , 
                 
                   ( 
                   
                     
                       n 
                       2 
                     
                     + 
                     
                       n 
                       1 
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     
                       - 
                       
                         n 
                         2 
                       
                     
                     + 
                     
                       n 
                       1 
                     
                     - 
                     
                       n 
                       2 
                     
                     - 
                     
                       n 
                       1 
                     
                   
                 
                  
                 
                   I 
                   
                     
                       n 
                       1 
                     
                     , 
                     
                       - 
                       
                         n 
                         2 
                       
                     
                     , 
                     
                       - 
                       
                         ( 
                         
                           
                             n 
                             2 
                           
                           + 
                           
                             n 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       
                         - 
                         2 
                       
                        
                       
                           
                       
                        
                       
                         n 
                         2 
                       
                     
                   
                    
                   
                     I 
                     
                       
                         n 
                         1 
                       
                       , 
                       
                         - 
                         
                           n 
                           2 
                         
                       
                       , 
                       
                         - 
                         
                           ( 
                           
                             
                               n 
                               2 
                             
                             + 
                             
                               n 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       I 
                       
                         
                           n 
                           1 
                         
                         , 
                         
                           - 
                           
                             n 
                             2 
                           
                         
                         , 
                         
                           - 
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                               + 
                               
                                 n 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                     ⇒ 
                     
                       
 
                     
                      
                     
                       
                         
                           
                             8 
                              
                             π 
                           
                           
                             
                               ( 
                               
                                 - 
                                 
                                   n 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 n 
                                 2 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     n 
                                     2 
                                   
                                   + 
                                   
                                     n 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                              
                             φ 
                           
                         
                          
                         
                           I 
                           
                             
                               - 
                               
                                 n 
                                 1 
                               
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               
                                 n 
                                 2 
                               
                               + 
                               
                                 n 
                                 1 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           
                             8 
                              
                             π 
                           
                           
                             
                               ( 
                               
                                 n 
                                 1 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 - 
                                 
                                   n 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             
                               - 
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       n 
                                       2 
                                     
                                     + 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             φ 
                           
                         
                          
                         
                           I 
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               - 
                               
                                 n 
                                 2 
                               
                             
                             , 
                             
                               - 
                               
                                 ( 
                                 
                                   
                                     n 
                                     2 
                                   
                                   + 
                                   
                                     n 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             8 
                              
                             π 
                           
                           
                             
                               n 
                               1 
                             
                              
                             
                               n 
                               2 
                             
                           
                         
                       
                        
                       
                         ( 
                         
                           
                              
                             
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       n 
                                       2 
                                     
                                     + 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               φ 
                             
                           
                           + 
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       
                                         n 
                                         2 
                                       
                                       + 
                                       
                                         n 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               φ 
                             
                           
                         
                         ) 
                       
                        
                       
                         I 
                         
                           
                             - 
                             
                               n 
                               1 
                             
                           
                           , 
                           
                             n 
                             2 
                           
                           , 
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                               + 
                               
                                 n 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           - 
                           
                             
                               16 
                                
                               π 
                             
                             
                               
                                 n 
                                 1 
                               
                                
                               
                                 n 
                                 2 
                               
                             
                           
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                               + 
                               
                                 n 
                                 1 
                               
                             
                             ) 
                           
                         
                          
                         
                           I 
                           
                             
                               - 
                               
                                 n 
                                 1 
                               
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             
                               ( 
                               
                                 
                                   n 
                                   2 
                                 
                                 + 
                                 
                                   n 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             
                               16 
                                
                               π 
                             
                             
                               
                                 n 
                                 1 
                               
                                
                               
                                 n 
                                 2 
                               
                             
                           
                            
                           cos 
                            
                           
                             ( 
                             
                               
                                 n 
                                 2 
                               
                               + 
                               
                                 n 
                                 1 
                               
                             
                             ) 
                           
                            
                           
                             I 
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 
                                   n 
                                   2 
                                 
                                 + 
                                 
                                   n 
                                   1 
                                 
                               
                             
                           
                         
                          
                         
                           
 
                         
                          
                         
                             
                         
                         ⇒ 
                       
                     
                   
                 
               
             
           
         
       
     
         [0230]    We can therefore replace the [−∞ to ∞] summations by [1 to ∞] and arrive at: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       BF 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     N 
                      
                     
                       { 
                       
                         2 
                          
                         
                           π 
                           3 
                         
                          
                         
                           
                             I 
                             
                               0 
                               , 
                               0 
                               , 
                               0 
                             
                           
                           ++ 
                         
                          
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   2 
                                 
                                 = 
                                 
                                   1 
                                    
                                   
                                     ( 
                                     odd 
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 
                                   n 
                                   1 
                                 
                                 = 
                                 
                                   1 
                                    
                                   
                                     ( 
                                     odd 
                                     ) 
                                   
                                 
                               
                             
                             ∞ 
                           
                            
                           
                             
                               
                                 16 
                                  
                                 π 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   n 
                                   2 
                                 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               n 
                                               2 
                                             
                                             - 
                                             
                                               n 
                                               1 
                                             
                                           
                                           ) 
                                         
                                          
                                         φ 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         
                                           n 
                                           2 
                                         
                                         - 
                                         
                                           n 
                                           1 
                                         
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               n 
                                               2 
                                             
                                             + 
                                             
                                               n 
                                               1 
                                             
                                           
                                           ) 
                                         
                                          
                                         φ 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         n 
                                         1 
                                       
                                       , 
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         
                                           n 
                                           2 
                                         
                                         + 
                                         
                                           n 
                                           1 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
       ANNEX B 
     Diffraction Tomography Algorithm for the Limited View Hemi-Spherical Aperture 
       [0231]    A new derivation of a three-dimensional DT based on a three-dimensional beamforming (BF) algorithm is discussed, as an alternative approach to standard DT algorithms such as the filtered backpropagation method 15 .  15  Devaney, A. J. 1982, “A filtered backpropagation algorithm for diffraction tomography”, Ultrason. Imaging 4, 336-350. 
         [0232]    We assume that the scattering problem is described by a scalar wave field, ψ, solution to 
         [0000]        H ψ( r,k   0   {circumflex over (r)}   0 ,ω)=− O ( r ,ω)ψ( r,k   0   {circumflex over (r)}   0 ,ω)  (1)
 
         [0000]    where H is the Helmholtz operator (∇ 2 +k 0   2 ), k 0  is the background wavenumber (2π/λ), {circumflex over (r)} 0  specifies the direction of an incident plane wave that illuminates the object and ω is the angular frequency. The unit vector {circumflex over (r)} 0  is defined by the angles θ t  and φ t  of a spherical coordinate systemError! Reference source not found. 
         [0233]    The object is described by the so-called Object Function that depends on the type of wave field used to probe the object: for electromagnetic wave sensing, it is related to the index of refraction 16 , n(r,ω), through the relation O(r)=k 2   0 [n 2 (r,ω)−1], and for acoustic waves, it is linked to the speed of sound and the attenuation coefficient 17 . In particular, for a lossless object 
         [0000]    
       
         
           
             
               
                 
                   
                     o 
                      
                     
                       ( 
                       
                         r 
                         , 
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       k 
                       0 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 c 
                                 0 
                               
                               
                                 c 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     , 
                                     ω 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         1 
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where c 0  is the sound speed of the homogeneous background in which the object is immersed and c(r,ω) is the local sound speed inside the object. The dependence of the object function on ω is because of dispersion and energy dissipation phenomena. The analysis performed in the rest of this section will consider monochromatic wave fields; therefore, the explicit dependence on ω is omitted.  16  Born, M. &amp; Wolf, E. 1999 Principles of optics. Cambridge, UK: Cambridge University Press. 17  Kak, A. C. &amp; Slaney, M. 1988 Principles of computerized tomographic imaging. New York, N.Y.: IEEE Press. 
       Three-Dimensional Beamforming Algorithm Over a Semi-Sphere 
       [0234]    Let us assume that the scattering amplitude, ƒ(θ r , θ t , φ r , φ t ), can be measured as a continuous function of the illumination and detection directions, i.e. θ r , θ t ε[0,π] and φ r , φ t ε[0,π] for a semi-sphere, (note that for a full sphere φ r , φ t ε[0,2π]), these angles being the receive and transmit directions in a spherical coordinate system respectively. In principle, this could be achieved with the semi-spherical array of transceivers that surrounds the object. 
         [0235]    Standard BF produces the image of an object at a point, z, of the image space by focusing an incident beam at r and z in the object space. The resulting scattered field is subsequently phase shifted and integrated over the aperture of the array, so that only the contributions to the scattered field from the focal point are added coherently. This two-step process is obtained by means of the BF functional 
         [0000]      ℑ BF =∫ 0   π   dφ   r ∫ 0   π   dθ   r  sin θ r ∫ 0   π   dφ   t ∫ 0   π   dθ   t  sin θ t ×exp[ ik   0   û (θ r ,φ r )· z ]ƒ(θ r ,θ t ,φ r ,φ t )exp[ ik   0   û (θ t ,φ t )· z]   (3)
 
         [0000]    where û is the unit vector associated with the angles θ and φ. As discussed by 18  for the two-dimensional case, the second exponential in equation (III) represents focusing in transmission, whereas the first corresponds to the focusing of the received scattered field. The point spread function (PSF) associated with the functional (III) can be obtained by considering the image of a point scatterer at position r. In this case, the free-scattering amplitude is 
         [0000]      ƒ free (θ r ,θ t ,φ r ,φ t )=exp{− ik   0   [û (θ t ,φ t )+ û (θ r ,φ r )]· r}   (4)
 
         [0000]      and the PSF reads 
         [0000]        h   BF =∫ 0   π   dφ   r ∫ 0   π   dθ   r  sin θ r ∫ 0   π   dφ   t ∫ 0   π   dθ   t  sin θ t ×exp{ ik   0   û (θ r ,φ r )·[ z−r ]}exp{ ik   0   û (θ t ,φ t )·[ z−r]}   (5)
 
         [0000]      18  Simonetti, F. &amp; Huang, L. 2008, “From beamforming to diffraction tomography”, J. Appl. Phys. 103, 103 110. 
         [0236]    Spherical waves expansion: 
         [0000]    
       
         
           
             
               exp 
                
               
                 { 
                 
                    
                    
                   
                       
                   
                    
                   
                     k 
                     0 
                   
                    
                   
                     
                       
                         u 
                         ^ 
                       
                        
                       
                         ( 
                         
                           
                             θ 
                             r 
                           
                           , 
                           
                             φ 
                             r 
                           
                         
                         ) 
                       
                     
                     · 
                     
                       [ 
                       
                         z 
                         - 
                         r 
                       
                       ] 
                     
                   
                 
                 } 
               
             
             = 
             
               
                 ∑ 
                 
                   l 
                   = 
                   0 
                 
                 ∞ 
               
                
               
                   
               
                
               
                 
                   
                     i 
                     l 
                   
                    
                   
                     ( 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         l 
                       
                       + 
                       1 
                     
                     ) 
                   
                 
                  
                 
                   
                     j 
                     l 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                     ) 
                   
                 
                  
                 
                   
                     P 
                     l 
                   
                    
                   
                     [ 
                     
                       cos 
                        
                       
                         ( 
                         
                           angle 
                            
                           
                             ( 
                             
                               
                                 
                                   u 
                                   ^ 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       r 
                                     
                                     , 
                                     
                                       φ 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
         [0000]    where j l  is the spherical Bessel function of the order l and P l  are Legendre Polynomials. 
         [0000]    
       
         
           
             
               
                 
                   ∫ 
                   0 
                   π 
                 
                  
                 
                     
                 
                  
                 
                   
                      
                     
                       φ 
                       r 
                     
                   
                    
                   
                     
                       ∫ 
                       0 
                       π 
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           θ 
                           r 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         r 
                       
                        
                       exp 
                        
                       
                         { 
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             0 
                           
                            
                           
                             
                               
                                 u 
                                 ^ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     θ 
                                     r 
                                   
                                   , 
                                   
                                     φ 
                                     r 
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               [ 
                               
                                 z 
                                 - 
                                 r 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               == 
               
                 
                   ∫ 
                   0 
                   π 
                 
                  
                 
                     
                 
                  
                 
                   
                      
                     
                       φ 
                       r 
                     
                   
                    
                   
                     
                       ∫ 
                       0 
                       π 
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           θ 
                           r 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         r 
                       
                        
                       
                         { 
                         
                           
                             
                               j 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   0 
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     - 
                                     r 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 1 
                               
                               ∞ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                   i 
                                   l 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       l 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   j 
                                   l 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   P 
                                   l 
                                 
                                  
                                 
                                   [ 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         angle 
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 u 
                                                 ^ 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     θ 
                                                     r 
                                                   
                                                   , 
                                                   
                                                     φ 
                                                     r 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             · 
                                             
                                               ( 
                                               
                                                 z 
                                                 - 
                                                 r 
                                               
                                               ) 
                                             
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
             
             = 
           
         
       
     
         [0237]    Define the angles representing z−r as θ′, φ′. The summation formula for the Spherical Harmonics is now intorduced 19 : 
         [0000]    
       
         
           
             
               
                 P 
                 l 
               
                
               
                 ( 
                 
                   cos 
                    
                   
                       
                   
                    
                   γ 
                 
                 ) 
               
             
             = 
             
               
                 
                   4 
                    
                   π 
                 
                 
                   
                     2 
                      
                     
                         
                     
                      
                     l 
                   
                   + 
                   1 
                 
               
                
               
                 
                   ∑ 
                   
                     m 
                     = 
                     
                       - 
                       1 
                     
                   
                   l 
                 
                  
                 
                     
                 
                  
                 
                   
                     
                       Y 
                       lm 
                       * 
                     
                      
                     
                       ( 
                       
                         
                           θ 
                           r 
                         
                         , 
                         
                           φ 
                           r 
                         
                       
                       ) 
                     
                   
                    
                   
                     
                       Y 
                       lm 
                     
                      
                     
                       ( 
                       
                         
                           θ 
                           ′ 
                         
                         , 
                         
                           φ 
                           ′ 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0000]    where
 
cos γ=cos θ cos θ′+sin θ sin θ′ cos(φ−φ′)  19  http://farside.ph.utexas.edu/teaching/jk1/lectures/node102.html
 
         [0238]    Therefore, the integration over θ r  and φ r  becomes: 
         [0000]    
       
         
           
             
               ∫ 
               0 
               π 
             
              
             
                 
             
              
             
               
                  
                 
                   φ 
                   r 
                 
               
                
               
                 
                   ∫ 
                   0 
                   π 
                 
                  
                 
                     
                 
                  
                 
                   
                      
                     
                       θ 
                       r 
                     
                   
                    
                   sin 
                    
                   
                       
                   
                    
                   
                     θ 
                     r 
                   
                    
                   
                     
                       Y 
                       lm 
                       * 
                     
                      
                     
                       ( 
                       
                         
                           θ 
                           r 
                         
                         , 
                         
                           φ 
                           r 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Y 
                 lm 
                 * 
               
                
               
                 ( 
                 
                   
                     θ 
                     r 
                   
                   , 
                   
                     φ 
                     r 
                   
                 
                 ) 
               
             
             ≡ 
             
               { 
               
                 
                   
                     
                       
                         K 
                         l 
                         m 
                       
                        
                       
                         
                           P 
                           l 
                           
                              
                             m 
                              
                           
                         
                          
                         
                           ( 
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               r 
                             
                           
                           ) 
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                                
                               m 
                                
                             
                              
                             
                               φ 
                               r 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         if 
                          
                         
                             
                         
                          
                         m 
                       
                       ≥ 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         K 
                         l 
                         m 
                       
                        
                       
                         
                           P 
                           l 
                           
                              
                             m 
                              
                           
                         
                          
                         
                           ( 
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               r 
                             
                           
                           ) 
                         
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                                
                               m 
                                
                             
                              
                             
                               φ 
                               r 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         if 
                          
                         
                             
                         
                          
                         m 
                       
                       &lt; 
                       0 
                     
                   
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               K 
               l 
               m 
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         l 
                       
                       + 
                       1 
                     
                     ) 
                   
                    
                   
                     
                       ( 
                       
                         l 
                         - 
                         m 
                       
                       ) 
                     
                     ! 
                   
                 
                 
                   4 
                    
                   
                     
                       π 
                        
                       
                         ( 
                         
                           l 
                           + 
                           m 
                         
                         ) 
                       
                     
                     ! 
                   
                 
               
             
           
         
       
     
         [0239]    As is well known, from the above definition it follows that the spherical Harmonics are separable in θ r , φ r . 
         [0240]    We now define the following integrals 20    
         [0000]    
       
         
           
             
               
                 ∫ 
                 
                   θ 
                   - 
                 
                 
                   θ 
                   + 
                 
               
                
               
                 
                   
                     P 
                     l 
                     
                        
                       m 
                        
                     
                   
                    
                   
                     ( 
                     
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     ) 
                   
                 
                  
                 sin 
                  
                 
                     
                 
                  
                 θ 
                  
                 
                     
                 
                  
                 
                    
                   θ 
                 
               
             
             = 
             
               
                 
                   ∫ 
                   
                     z 
                     - 
                   
                   
                     z 
                     + 
                   
                 
                  
                 
                   
                     
                       P 
                       l 
                       
                          
                         m 
                          
                       
                     
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                    
                   
                       
                   
                    
                   
                      
                     z 
                   
                 
               
               = 
               
                 
                   
                     P 
                     ^ 
                   
                   l 
                   
                      
                     m 
                      
                   
                 
                  
                 
                   ( 
                   
                     
                       θ 
                       - 
                     
                     , 
                     
                       θ 
                       + 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where z=cos θ, and z ± =cos θ ± .  20  W. Jarosz, N. Carr &amp; H. W. Jensen, “Importance Sampling Spherical Harmonics”, Journal compilation, 2008, The Eurographics Association and Blackwell Publishing Ltd. 
         [0241]    We also define 
         [0000]    
       
         
           
             
               
                 Φ 
                 m 
               
                
               
                 ( 
                 φ 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           cos 
                            
                           
                             ( 
                             
                               m 
                                
                               
                                   
                               
                                
                               φ 
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             m 
                           
                           ≥ 
                           0 
                         
                       
                     
                     
                       
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                  
                                 m 
                                  
                               
                                
                               φ 
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             m 
                           
                           &lt; 
                           0 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         Φ 
                         ^ 
                       
                       m 
                     
                      
                     
                       ( 
                       φ 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       
                         
                           Φ 
                           m 
                         
                          
                         
                           ( 
                           φ 
                           ) 
                         
                       
                        
                       
                          
                         φ 
                       
                     
                   
                   = 
                   
                     
                       1 
                       m 
                     
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       φ 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 
                                   
                                     if 
                                      
                                     
                                         
                                     
                                      
                                     m 
                                   
                                   ≥ 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       φ 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 
                                   
                                     if 
                                      
                                     
                                         
                                     
                                      
                                     m 
                                   
                                   &lt; 
                                   0 
                                 
                               
                             
                           
                            
                           
                             
 
                           
                            
                           
                             
                               
                                 Φ 
                                 ^ 
                               
                               m 
                             
                              
                             
                               ( 
                               
                                 
                                   φ 
                                   - 
                                 
                                 , 
                                 
                                   φ 
                                   + 
                                 
                               
                               ) 
                             
                           
                         
                         = 
                         
                           
                             ∫ 
                             
                               φ 
                               - 
                             
                             
                               φ 
                               + 
                             
                           
                            
                           
                             
                               
                                 Φ 
                                 m 
                               
                                
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                              
                             
                                 
                             
                              
                             
                                
                               φ 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0242]    And with {circumflex over (φ)} m  and {circumflex over (P)} l   |m|  we define: 
         [0000]        Ŷ   lm (θ − ,θ + ,φ − ,φ + )= K   l   m   {circumflex over (P)}   l   |m| (θ − ,θ + ){circumflex over (φ)} m (φ − ,φ + )
 
         [0243]    We now go back to the beamforming terms and look at: 
         [0000]    
       
         
           
             
               
                 g 
                 r 
               
                
               
                 ( 
                 
                   z 
                   - 
                   r 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     ∫ 
                     0 
                     π 
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         r 
                       
                     
                      
                     
                       
                         ∫ 
                         0 
                         π 
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           
                             θ 
                             r 
                           
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           θ 
                           r 
                         
                          
                         exp 
                          
                         
                           { 
                           
                              
                              
                             
                                 
                             
                              
                             
                               k 
                               0 
                             
                              
                             
                               
                                 
                                   u 
                                   ^ 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       r 
                                     
                                     , 
                                     
                                       φ 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 [ 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ] 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
                 == 
                 
                   
                     ∫ 
                     0 
                     π 
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       
                         φ 
                         r 
                       
                     
                      
                     
                       
                         ∫ 
                         0 
                         π 
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           
                             θ 
                             r 
                           
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           θ 
                           r 
                         
                          
                         
                           { 
                           
                             
                               
                                 j 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   l 
                                   = 
                                   1 
                                 
                                 ∞ 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     i 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         l 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     j 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         0 
                                       
                                        
                                       
                                          
                                         
                                           z 
                                           - 
                                           r 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     4 
                                      
                                     π 
                                   
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       l 
                                     
                                     + 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       
                                         - 
                                         1 
                                       
                                     
                                     l 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         Y 
                                         lm 
                                         * 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             r 
                                           
                                           , 
                                           
                                             φ 
                                             r 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         Y 
                                         lm 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             ′ 
                                           
                                           , 
                                           
                                             φ 
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
                 == 
                 
                   
                     2 
                      
                     π 
                      
                     
                         
                     
                      
                     
                       
                         j 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         l 
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           i 
                           l 
                         
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               l 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                        
                       
                         
                           j 
                           l 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               0 
                             
                              
                             
                                
                               
                                 z 
                                 - 
                                 r 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           4 
                            
                           π 
                         
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             l 
                           
                           + 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             
                               - 
                               1 
                             
                           
                           l 
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               
                                 Y 
                                 ^ 
                               
                               lm 
                               * 
                             
                              
                             
                               ( 
                               
                                 0 
                                 , 
                                 π 
                                 , 
                                 0 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                            
                           
                             
                               Y 
                               lm 
                             
                              
                             
                               ( 
                               
                                 
                                   θ 
                                   ′ 
                                 
                                 , 
                                 
                                   φ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               = 
               
                 
                   ∑ 
                   
                     l 
                     = 
                     0 
                   
                   ∞ 
                 
                  
                 
                     
                 
                  
                 
                   
                     ∑ 
                     
                       m 
                       = 
                       
                         - 
                         l 
                       
                     
                     l 
                   
                    
                   
                       
                   
                    
                   
                     
                       A 
                       l 
                       m 
                     
                      
                     
                       
                         j 
                         l 
                       
                        
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                            
                           
                              
                             
                               z 
                               - 
                               r 
                             
                              
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         Y 
                         lm 
                       
                        
                       
                         ( 
                         
                           
                             θ 
                             ′ 
                           
                           , 
                           
                             φ 
                             ′ 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               A 
               l 
               m 
             
             = 
             
               
                 
                   
                     i 
                     l 
                   
                    
                   
                     ( 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         l 
                       
                       + 
                       1 
                     
                     ) 
                   
                 
                  
                 
                   
                     4 
                      
                     π 
                   
                   
                     
                       2 
                        
                       
                           
                       
                        
                       l 
                     
                     + 
                     1 
                   
                 
                  
                 
                   
                     
                       Y 
                       ^ 
                     
                     lm 
                     * 
                   
                    
                   
                     ( 
                     
                       0 
                       , 
                       π 
                       , 
                       0 
                       , 
                       n 
                     
                     ) 
                   
                 
               
               = 
               
                 4 
                  
                 π 
                  
                 
                     
                 
                  
                 
                   i 
                   l 
                 
                  
                 
                   
                     
                       Y 
                       ^ 
                     
                     lm 
                     * 
                   
                    
                   
                     ( 
                     
                       0 
                       , 
                       π 
                       , 
                       0 
                       , 
                       n 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0244]    The same result is obtained for the transmit angles quantity g t (z−r), so we now calculate the three-dimensional Fourier transform H BF (k) of h BF (z−r) given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         h 
                         BF 
                       
                        
                       
                         ( 
                         
                           z 
                           - 
                           r 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             g 
                             r 
                           
                            
                           
                             ( 
                             
                               z 
                               - 
                               r 
                             
                             ) 
                           
                         
                          
                         
                           
                             g 
                             t 
                           
                            
                           
                             ( 
                             
                               z 
                               - 
                               r 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               ∞ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   
                                     - 
                                     l 
                                   
                                 
                                 l 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   A 
                                   l 
                                   m 
                                 
                                  
                                 
                                   
                                     j 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         0 
                                       
                                        
                                       
                                          
                                         
                                           z 
                                           - 
                                           r 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     Y 
                                     lm 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         ′ 
                                       
                                       , 
                                       
                                         φ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         × 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 
                                   l 
                                   ′ 
                                 
                                 = 
                                 0 
                               
                               ∞ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     m 
                                     ′ 
                                   
                                   = 
                                   
                                     - 
                                     
                                       l 
                                       ′ 
                                     
                                   
                                 
                                 
                                   l 
                                   ′ 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   A 
                                   
                                     l 
                                     ′ 
                                   
                                   
                                     m 
                                     ′ 
                                   
                                 
                                  
                                 
                                   
                                     j 
                                     
                                       l 
                                       ′ 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         0 
                                       
                                        
                                       
                                          
                                         
                                           z 
                                           - 
                                           r 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     Y 
                                     
                                       
                                         l 
                                         ′ 
                                       
                                        
                                       
                                         m 
                                         ′ 
                                       
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         ′ 
                                       
                                       , 
                                       
                                         φ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         H 
                         BF 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           
                             
                                
                               3 
                             
                              
                             
                               
                                 rg 
                                 r 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             
                               g 
                               t 
                             
                              
                             
                               ( 
                               
                                 z 
                                 - 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 · 
                                 
                                   [ 
                                   
                                     z 
                                     - 
                                     r 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       == 
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           ∞ 
                         
                          
                         
                             
                         
                          
                         
                           
                             
                                
                               3 
                             
                              
                             r 
                           
                            
                           
                               
                           
                            
                           
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   · 
                                   
                                     [ 
                                     
                                       z 
                                       - 
                                       r 
                                     
                                     ] 
                                   
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     l 
                                     = 
                                     0 
                                   
                                   ∞ 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       
                                         - 
                                         l 
                                       
                                     
                                     l 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       A 
                                       l 
                                       m 
                                     
                                      
                                     
                                       
                                         j 
                                         l 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                              
                                             
                                               z 
                                               - 
                                               r 
                                             
                                              
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         Y 
                                         lm 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             ′ 
                                           
                                           , 
                                           
                                             φ 
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           × 
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   
                                     l 
                                     ′ 
                                   
                                   = 
                                   0 
                                 
                                 ∞ 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     
                                       m 
                                       ′ 
                                     
                                     = 
                                     
                                       - 
                                       
                                         l 
                                         ′ 
                                       
                                     
                                   
                                   
                                     l 
                                     ′ 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     A 
                                     
                                       l 
                                       ′ 
                                     
                                     
                                       m 
                                       ′ 
                                     
                                   
                                    
                                   
                                     
                                       j 
                                       
                                         l 
                                         ′ 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           0 
                                         
                                          
                                         
                                            
                                           
                                             z 
                                             - 
                                             r 
                                           
                                            
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       Y 
                                       
                                         
                                           l 
                                           ′ 
                                         
                                          
                                         
                                           m 
                                           ′ 
                                         
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           ′ 
                                         
                                         , 
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0245]    We use the expansion in spherical waves once again: 
         [0000]      exp{− ik·[z−r]}=Σ   l=0   ∞   i   l (2 l+ 1) j   l ( k   0   |z−r |) P   l [cos(angle( k· ( z−r )))]
 
         [0246]    Denoting by θ,φ the angles of k and as before θ′,φ′ are angles of −r, (addition theorem): 
         [0000]    
       
         
           
             
               
                 
                   
                     exp 
                      
                     
                       { 
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         
                           k 
                           · 
                           
                             [ 
                             
                               z 
                               - 
                               r 
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           
                             l 
                             ″ 
                           
                           = 
                           0 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             i 
                             
                               l 
                               ″ 
                             
                           
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   l 
                                   ″ 
                                 
                               
                               + 
                               1 
                             
                             ) 
                           
                         
                          
                         
                           
                             j 
                             
                               l 
                               ″ 
                             
                           
                            
                           
                             ( 
                             
                               
                                  
                                 k 
                                  
                               
                                
                               
                                  
                                 
                                   z 
                                   - 
                                   r 
                                 
                                  
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             4 
                              
                             π 
                           
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 l 
                                 ″ 
                               
                             
                             + 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               
                                 m 
                                 ″ 
                               
                               = 
                               
                                 - 
                                 
                                   l 
                                   ″ 
                                 
                               
                             
                             
                               l 
                               ″ 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 Y 
                                 
                                   
                                     l 
                                     ″ 
                                   
                                    
                                   
                                     m 
                                     ″ 
                                   
                                 
                                 * 
                               
                                
                               
                                 ( 
                                 
                                   θ 
                                   , 
                                   φ 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 Y 
                                 
                                   
                                     l 
                                     ″ 
                                   
                                    
                                   
                                     m 
                                     ″ 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     θ 
                                     ′ 
                                   
                                   , 
                                   
                                     φ 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
 
                             
                              
                             
                               H 
                               BF 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                          
                         
                           
                             r 
                             2 
                           
                            
                           
                               
                           
                            
                           
                              
                             r 
                           
                            
                           
                             
                               ∫ 
                               0 
                               π 
                             
                              
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 ′ 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   θ 
                                   ′ 
                                 
                               
                                
                               
                                 
                                   ∫ 
                                   0 
                                   
                                     2 
                                      
                                     π 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     
                                       
                                         φ 
                                         ′ 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             ∑ 
                                             
                                               
                                                 l 
                                                 ″ 
                                               
                                               = 
                                               0 
                                             
                                             ∞ 
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               
                                                 i 
                                                 
                                                   l 
                                                   ″ 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       l 
                                                       ″ 
                                                     
                                                   
                                                   + 
                                                   1 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               
                                                 j 
                                                 
                                                   l 
                                                   ″ 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                      
                                                     k 
                                                      
                                                   
                                                    
                                                   
                                                      
                                                     
                                                       z 
                                                       - 
                                                       r 
                                                     
                                                      
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               
                                                 4 
                                                  
                                                 π 
                                               
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     l 
                                                     ″ 
                                                   
                                                 
                                                 + 
                                                 1 
                                               
                                             
                                              
                                             
                                               
                                                 ∑ 
                                                 
                                                   
                                                     m 
                                                     ″ 
                                                   
                                                   = 
                                                   
                                                     - 
                                                     
                                                       l 
                                                       ″ 
                                                     
                                                   
                                                 
                                                 
                                                   l 
                                                   ″ 
                                                 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 
                                                   
                                                     Y 
                                                     
                                                       
                                                         l 
                                                         ″ 
                                                       
                                                        
                                                       
                                                         m 
                                                         ″ 
                                                       
                                                     
                                                     * 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       θ 
                                                       , 
                                                       φ 
                                                     
                                                     ) 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     Y 
                                                     
                                                       
                                                         l 
                                                         ″ 
                                                       
                                                        
                                                       
                                                         m 
                                                         ″ 
                                                       
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         θ 
                                                         ′ 
                                                       
                                                       , 
                                                       
                                                         φ 
                                                         ′ 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   × 
                                   
                                     ( 
                                     
                                       
                                         ∑ 
                                         
                                           l 
                                           = 
                                           0 
                                         
                                         ∞ 
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           ∑ 
                                           
                                             m 
                                             = 
                                             
                                               - 
                                               l 
                                             
                                           
                                           l 
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             A 
                                             l 
                                             m 
                                           
                                            
                                           
                                             
                                               j 
                                               l 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   0 
                                                 
                                                  
                                                 
                                                    
                                                   
                                                     z 
                                                     - 
                                                     r 
                                                   
                                                    
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                               Y 
                                               lm 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   θ 
                                                   ′ 
                                                 
                                                 , 
                                                 
                                                   φ 
                                                   ′ 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       
                                         ∑ 
                                         
                                           
                                             l 
                                             ′ 
                                           
                                           = 
                                           0 
                                         
                                         ∞ 
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           ∑ 
                                           
                                             
                                               m 
                                               ′ 
                                             
                                             = 
                                             
                                               - 
                                               
                                                 l 
                                                 ′ 
                                               
                                             
                                           
                                           
                                             l 
                                             ′ 
                                           
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             A 
                                             
                                               l 
                                               ′ 
                                             
                                             
                                               m 
                                               ′ 
                                             
                                           
                                            
                                           
                                             
                                               j 
                                               
                                                 l 
                                                 ′ 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   0 
                                                 
                                                  
                                                 
                                                    
                                                   
                                                     z 
                                                     - 
                                                     r 
                                                   
                                                    
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                               Y 
                                               
                                                 
                                                   l 
                                                   ′ 
                                                 
                                                  
                                                 
                                                   m 
                                                   ′ 
                                                 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   θ 
                                                   ′ 
                                                 
                                                 , 
                                                 
                                                   φ 
                                                   ′ 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     
 
                                   
                                    
                                   
                                     
                                       H 
                                       BF 
                                     
                                      
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             l 
                             , 
                             m 
                             , 
                             
                               l 
                               ′ 
                             
                             , 
                             
                               m 
                               ′ 
                             
                             , 
                             
                               l 
                               ″ 
                             
                             , 
                             
                               m 
                               ″ 
                             
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             B 
                             
                               l 
                               ″ 
                             
                           
                            
                           
                             C 
                             
                               l 
                               , 
                               m 
                               , 
                               
                                 l 
                                 ′ 
                               
                               , 
                               
                                 m 
                                 ′ 
                               
                               , 
                               
                                 l 
                                 ″ 
                               
                               , 
                               
                                 m 
                                 ″ 
                               
                             
                           
                            
                           
                             
                               Y 
                               
                                 
                                   l 
                                   ″ 
                                 
                                  
                                 
                                   m 
                                   ″ 
                                 
                               
                               * 
                             
                              
                             
                               ( 
                               
                                 θ 
                                 , 
                                 φ 
                               
                               ) 
                             
                           
                           × 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                              
                             
                               
                                 r 
                                 2 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 r 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   j 
                                   
                                     l 
                                     ″ 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                        
                                       k 
                                        
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   j 
                                   
                                     l 
                                     ′ 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   j 
                                   l 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       0 
                                     
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         r 
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where C l,m,l′,m′,l″,m″ =∫ 0   π  sin θ′ dθ′∫ 0   2π dφ′Y lm (θ′,φ′)Y l′m′ (θ′,φ′)Y l″m″ (θ′,φ′) and B l″ =4πi l″ . 
         [0247]    We now focus on the integral of a product of 3 spherical Bessel functions 21 : 
         [0000]    
       
         
           
             
               I 
                
               
                 ( 
                 
                   
                     λ 
                     1 
                   
                   , 
                   
                     λ 
                     2 
                   
                   , 
                   
                     
                       λ 
                       3 
                     
                     ; 
                     
                       k 
                       1 
                     
                   
                   , 
                   
                     k 
                     2 
                   
                   , 
                   
                     k 
                     3 
                   
                 
                 ) 
               
             
             ≡ 
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                   r 
                   2 
                 
                  
                 
                     
                 
                  
                 
                    
                   r 
                 
                  
                 
                     
                 
                  
                 
                   
                     j 
                     
                       λ 
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         k 
                         1 
                       
                        
                       r 
                     
                     ) 
                   
                 
                  
                 
                   
                     j 
                     
                       λ 
                       2 
                     
                   
                   ( 
                   
                     
                       ( 
                       
                         
                           k 
                           2 
                         
                          
                         r 
                       
                       ) 
                     
                      
                     
                       
                         j 
                         
                           λ 
                           3 
                         
                       
                       ( 
                       
                         
                           
                             ( 
                             
                               
                                 k 
                                 3 
                               
                                
                               r 
                             
                             ) 
                           
                            
                           
                             
 
                           
                            
                           
                             I 
                              
                             
                               ( 
                               
                                 
                                   λ 
                                   1 
                                 
                                 , 
                                 
                                   λ 
                                   2 
                                 
                                 , 
                                 
                                   
                                     λ 
                                     3 
                                   
                                   ; 
                                   
                                     k 
                                     1 
                                   
                                 
                                 , 
                                 
                                   k 
                                   2 
                                 
                                 , 
                                 
                                   k 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         = 
                         
                           
                             
                               πβ 
                                
                               
                                 ( 
                                 Δ 
                                 ) 
                               
                             
                             
                               4 
                                
                               
                                   
                               
                                
                               
                                 k 
                                 1 
                               
                                
                               
                                 k 
                                 2 
                               
                                
                               
                                 k 
                                 3 
                               
                             
                           
                            
                           
                             
                               
                                 i 
                                 
                                   
                                     λ 
                                     1 
                                   
                                   + 
                                   
                                     λ 
                                     2 
                                   
                                   - 
                                   
                                     λ 
                                     3 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       λ 
                                       3 
                                     
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             
                               1 
                               2 
                             
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   
                                     k 
                                     1 
                                   
                                   
                                     k 
                                     3 
                                   
                                 
                                 ) 
                               
                               
                                 λ 
                                 3 
                               
                             
                             · 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         λ 
                                         1 
                                       
                                     
                                     
                                       
                                         λ 
                                         2 
                                       
                                     
                                     
                                       
                                         λ 
                                         3 
                                       
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 L 
                                 = 
                                 0 
                               
                               
                                 λ 
                                 3 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           2 
                                            
                                           
                                             λ 
                                             3 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           L 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 
                                   1 
                                   2 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       k 
                                       2 
                                     
                                     
                                       k 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 L 
                               
                                
                               
                                 
                                   ∑ 
                                   l 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         l 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               λ 
                                               1 
                                             
                                           
                                           
                                             
                                               
                                                 λ 
                                                 3 
                                               
                                               - 
                                               L 
                                             
                                           
                                           
                                             l 
                                           
                                         
                                         
                                           
                                             0 
                                           
                                           
                                             0 
                                           
                                           
                                             0 
                                           
                                         
                                       
                                       ) 
                                     
                                     · 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               λ 
                                               2 
                                             
                                           
                                           
                                             L 
                                           
                                           
                                             l 
                                           
                                         
                                         
                                           
                                             0 
                                           
                                           
                                             0 
                                           
                                           
                                             0 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     { 
                                     
                                       
                                         
                                           
                                             λ 
                                             1 
                                           
                                         
                                         
                                           
                                             λ 
                                             2 
                                           
                                         
                                         
                                           
                                             λ 
                                             3 
                                           
                                         
                                       
                                       
                                         
                                           L 
                                         
                                         
                                           
                                             
                                               λ 
                                               3 
                                             
                                             - 
                                             L 
                                           
                                         
                                         
                                           l 
                                         
                                       
                                     
                                     } 
                                   
                                    
                                   
                                     
                                       P 
                                       l 
                                     
                                      
                                     
                                       ( 
                                       Δ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    where
 
|k 1 −k 2 |≦k 3 ≦k 1 +k 2 (closed triangle, angular momentum conservation)
 
         [0000]    
       
         
           
             Δ 
             = 
             
               
                 
                   k 
                   1 
                   2 
                 
                 + 
                 
                   k 
                   2 
                   2 
                 
                 - 
                 
                   k 
                   3 
                   2 
                 
               
               
                 2 
                  
                 
                   k 
                   1 
                 
                  
                 
                   k 
                   2 
                 
               
             
           
         
       
     
         [0000]    Δ lies between ±1 and is the cosine of the angle between {circumflex over (k)} 1  and {circumflex over (k)} 2  in the triangle formed by k 1 , k 2  and k 3 .  21  R Mehremt, 3 T Londergant and M H Macfarlanet, “Analytic expressions for integrals of products of spherical Bessel functions”, J. Phys. A: Math. Gen. 24(1991) 1435-1453. 
         [0248]    The quotation for I(λ 1 , λ 2 , λ 3 ; k 1 , k 2 , k 3 ) is valid for all real Δ, including values outside the limited range −1≦Δ≦1, with correct account taken of the jump discontinuities at Δ=±1, through the introduction of the function 
         [0000]      β(Δ)=∂(1−Δ)∂(1+Δ)
 
         [0000]    where ∂(y) is a modified step function: 
         [0000]    
       
         
           
             
               ϑ 
                
               
                 ( 
                 y 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     0 
                   
                   
                     
                       y 
                       &lt; 
                       0 
                     
                   
                 
                 
                   
                     
                       1 
                       2 
                     
                   
                   
                     
                       y 
                       = 
                       0 
                     
                   
                 
                 
                   
                     1 
                   
                   
                     
                       y 
                       &gt; 
                       0 
                     
                   
                 
               
             
           
         
       
     
         [0000]    
       
         
           
               
             
               ( 
               
                 
                   
                     
                       λ 
                       1 
                     
                   
                   
                     
                       λ 
                       2 
                     
                   
                   
                     
                       λ 
                       3 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    is the is the Wigner 3-j symbol 22  from which the angular momentum triangle is deduced, and recoupling of three angular momenta involves the 6-j symbol 
         [0000]    
       
         
           
             
               { 
               
                 
                   
                     
                       j 
                       1 
                     
                   
                   
                     
                       j 
                       2 
                     
                   
                   
                     
                       j 
                       3 
                     
                   
                 
                 
                   
                     
                       j 
                       4 
                     
                   
                   
                     
                       j 
                       5 
                     
                   
                   
                     
                       j 
                       6 
                     
                   
                 
               
               } 
             
             . 
           
         
       
     
         [0000]      22  Edmonds A R 1957 Angular Momentum in Quantum Mechanics (Princeton: Princeton University Academic Press) 
         [0249]    Our integral is: 
         [0000]    
       
         
           
             
               I 
                
               
                 ( 
                 
                   l 
                   , 
                   
                     l 
                     ′ 
                   
                   , 
                   
                     
                       l 
                       ″ 
                     
                     ; 
                     k 
                   
                   , 
                   
                     k 
                     0 
                   
                   , 
                   
                     k 
                     0 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                   r 
                   2 
                 
                  
                 
                    
                   
                     
                       rj 
                       
                         l 
                         ″ 
                       
                     
                      
                     
                       ( 
                       
                         
                            
                           k 
                            
                         
                          
                         
                            
                           
                             z 
                             - 
                             r 
                           
                            
                         
                       
                       ) 
                     
                   
                 
                  
                 
                   
                     j 
                     
                       l 
                       ′ 
                     
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                     ) 
                   
                 
                  
                 
                   
                     j 
                     l 
                   
                    
                   
                     ( 
                     
                       
                         k 
                         0 
                       
                        
                       
                          
                         
                           z 
                           - 
                           r 
                         
                          
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0250]    We therefore arrived at an analytic formula for H BF  of the semi-sphere, and in fact other limited view angles. Next we consider the value of Δ for our case: 
         [0251]    With k 1 =|k|, and k 2 =k 3 =k 0  we get: 
         [0000]    
       
         
           
             Δ 
             = 
             
               
                 
                   
                     
                        
                       k 
                        
                     
                     2 
                   
                   + 
                   
                     k 
                     0 
                     2 
                   
                   - 
                   
                     k 
                     0 
                     2 
                   
                 
                 
                   2 
                    
                   
                      
                     k 
                      
                   
                    
                   
                     k 
                     0 
                   
                 
               
               = 
               
                 
                    
                   k 
                    
                 
                 
                   2 
                    
                   
                     k 
                     0 
                   
                 
               
             
           
         
       
     
         [0252]    As −1≦Δ≦1, we get the low pass filtering: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       BF 
                     
                     = 
                     
                       
                         g 
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                        
                       
                         Π 
                          
                         
                           ( 
                           
                              
                             k 
                              
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       where 
                        
                       
                           
                       
                        
                       
                         Π 
                          
                         
                           ( 
                           
                              
                             k 
                              
                           
                           ) 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             1 
                           
                           
                             
                               
                                  
                                 k 
                                  
                               
                               &lt; 
                               
                                 2 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                  
                                 k 
                                  
                               
                               &gt; 
                               
                                 2 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0253]    The DT problem consists of reconstructing the function O(r) from a set of scattering experiments. For this purpose, it is convenient to introduce the representation of the object function in the spatial frequency domain, K-space, which is obtained by performing the three-dimensional Fourier transform of O(r) 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       O 
                       ~ 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                      
                     
                       
                         
                            
                           3 
                         
                          
                         r 
                       
                        
                       
                           
                       
                        
                       
                         O 
                          
                         
                           ( 
                           r 
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             k 
                             · 
                             r 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0254]    We now look at the beamforming image: 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     BF 
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                      
                     
                       
                          
                         
                           r 
                           1 
                         
                       
                        
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           ∞ 
                         
                          
                         
                           
                              
                             
                               r 
                               2 
                             
                           
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               ∞ 
                             
                              
                             
                               
                                  
                                 
                                   r 
                                   3 
                                 
                               
                                
                               
                                 O 
                                  
                                 
                                   ( 
                                   r 
                                   ) 
                                 
                               
                                
                               
                                 h 
                                  
                                 
                                   ( 
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    that in the spatial frequency domain reads 
         [0000]        I   BF ( k )= Õ ( k ) H   BF ( k )= g ( k ) Õ ( k )Π(| k |)  (11)
 
         [0255]    While DT over the entire sphere leads to the low-pass-filtered image, Õ(k)Π(|k|), the new BF algorithm introduces a distortion that is described by the additional filter g(k). As a result, the DT image can be obtained from the BF image by applying the filter 
         [0000]    
       
         
           
             1 
             
               g 
                
               
                 ( 
                 k 
                 ) 
               
             
           
         
       
     
         [0000]    to the BF image. Again, this is an alternative approach to other DT algorithms 23 .  23  Reference of footnote 3 
         [0256]    With no loss of generality we can choose the vector k parallel to the z axis, i.e θ=0 and cos θ=1. 
         [0257]    For this case, 
         [0000]    
       
         
           
             
               
                 Y 
                 
                   
                     l 
                     ″ 
                   
                    
                   
                     m 
                     ″ 
                   
                 
               
                
               
                 ( 
                 
                   0 
                   , 
                   φ 
                 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       
                         m 
                         ″ 
                       
                       = 
                       
                         0 
                          
                         
                             
                         
                          
                         for 
                          
                         
                             
                         
                          
                         all 
                          
                         
                             
                         
                          
                         
                           l 
                           ″ 
                         
                       
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         m 
                         ″ 
                       
                       ≠ 
                       
                         0 
                          
                         
                             
                         
                          
                         for 
                          
                         
                             
                         
                          
                         all 
                          
                         
                             
                         
                          
                         
                           l 
                           ″ 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    so the equation becomes independent of angles, i.e depends on |k|only: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       BF 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           l 
                           , 
                           m 
                           , 
                           
                             l 
                             ′ 
                           
                           , 
                           
                             m 
                             ′ 
                           
                           , 
                           
                             l 
                             ″ 
                           
                         
                       
                        
                       
                         
                           B 
                           
                             l 
                             ″ 
                           
                         
                          
                         
                           C 
                           
                             l 
                             , 
                             m 
                             , 
                             
                               l 
                               ′ 
                             
                             , 
                             
                               m 
                               ′ 
                             
                             , 
                             
                               l 
                               ″ 
                             
                             , 
                             0 
                           
                         
                         × 
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                            
                           
                             
                               r 
                               2 
                             
                              
                             
                                
                               r 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 j 
                                 
                                   l 
                                   ″ 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                      
                                     k 
                                      
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 j 
                                 
                                   i 
                                   ′ 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 j 
                                 l 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       - 
                                       r 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         H 
                         BF 
                       
                        
                       
                         ( 
                         
                            
                           k 
                            
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where again C l,m,l′,m′,l″,m″ =∫ 0   π  sin θ′ dθ′∫ 0   2π dφ′Y lm (θ′,φ′)Y l′m′ (θ′,φ′)Y l″m″ (θ′,φ′) and B l″ =4πi l″ . 
         [0258]    Therefore, the filter function ƒ(k) becomes a function of |k|only, ƒ(|k|). As 
         [0000]    
       
         
           
             
               Δ 
               = 
               
                 
                    
                   k 
                    
                 
                 
                   2 
                    
                   
                     k 
                     0 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    the filter ƒ(|k|) in this particular coordinate choice becomes a sum over legendre polynomials in 
         [0000]    
       
         
           
             
                
               k 
                
             
             
               2 
                
               
                 k 
                 0 
               
             
           
         
       
     
         [0000]    
       
         
           
             
               f 
                
               
                 ( 
                 
                    
                   k 
                    
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 n 
               
                
               
                 
                   M 
                   n 
                 
                  
                 
                   
                     P 
                     n 
                   
                    
                   
                     ( 
                     
                       
                          
                         k 
                          
                       
                       
                         2 
                          
                         
                           k 
                           0 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    
       
         
               
               
             
           
               
                   
               
               
                   n 
                 Legendre Polynomials 
           P   n          (   x   )       ;     x   =          k          2        k   0               
 
               
               
                   
               
             
             
               
                 0 
                 1 
               
               
                 1 
                 x 
               
               
                   
               
               
                 2 
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               3 
                                
                               
                                 x 
                                 2 
                               
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 3 
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               5 
                                
                               
                                 x 
                                 3 
                               
                             
                             - 
                             
                               3 
                                
                               x 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 4 
                 
                   
                     
                       
                         
                           1 
                           8 
                         
                          
                         
                           ( 
                           
                             
                               35 
                                
                               
                                 x 
                                 4 
                               
                             
                             - 
                             
                               30 
                                
                               
                                 x 
                                 2 
                               
                             
                             + 
                             3 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 . . . 
                 . . . 
               
               
                   
               
               
                 n 
                 
                   
                     
                       
                         
                           
                             P 
                             n 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             n 
                           
                            
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               n 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       n 
                                     
                                   
                                   
                                     
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         n 
                                         + 
                                         k 
                                       
                                     
                                   
                                   
                                     
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   ( 
                                   
                                     - 
                                     x 
                                   
                                   ) 
                                 
                                 k 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0259]    The summation over n denotes symbolically the multiple indices that need to be summed over. Note that the cooeficients M n  in the summation over n in the above equation for ƒ(|k|) are known, many of which vanish through, for example, the values of the 3-j and 6-j symbols of the multiple indices in this symbolically denoted summation.