best possible upper bounds for the first positive zeros of bessel functions the finite part it is shown here that the first three terms of the asymptotic expansion of jvk k 1 2 3 provide an upper bound for jvk in 0 v 10 hence a best possible upper bound lang and wong have shown that this is true also for 10 v when k 1 and 2 so that these best possible upper bounds hold in the entire interval 0 v in these cases we include supplementary comments on lower bounds in 0 v 10

an inequality for legendre polynomials the following inequality is established pn cos 1 4 16 n 1 2 4 sin4 1 0 n 1 2 where pn x denotes the legendre polynomial of degree n the relation p2n cos 4 2 q2n cos 1 4 16 n 1 2 4 sin4 1 n 1 2 on n1 n n 1 is proven where qn x denotes the legendre function of second kind cos n1 the largest zero of qn x and cos n n 1 cos n1 similarly we obtain the inequalities j0 x 1 4 16 x4 1 x 0 and j20 x y20 x 1 4 16 x4 1 x y1 where y1 0 893577 is the first positive zero of y0 x and j0 x y0 x denote the bessel functions of the first and second kind respectively the results of the present paper arise out of some problems of nuclear and particle physics

a series expansion for the first positive zero of the bessel functions it is shown that the first positive zerojv x of the bessel function j x is given by lp 2 v 1 1 2 1 v 1 7 v 1 2 49 v 1 3 8363 v 1 4 9 6 1152 276480 for 1 1 the bessel function j x has an infinite number of zeros and that all zeros are real watson 9 we denote the sth positive zero of jp x byj s several approximations asymptotic expansions or bounds for the zeros of bessel functions exist see 1 2 4 6 7 9 especially mcmahon s expansion for large zeros see abramowitz and stegun 1 olver s asymptotic expansion for large orders and olver s uniform asymptotic expansions see olver 6 are interesting formulas but unfortunately they are not applicable when s and v are small the purpose of this note is to give a series expansion forj 1 i when 1 v 0 2 cayley 3 noticed that graeffe s method for solving a polynomial equation can be applied for the efficient computation of 1 2r r r 1 2

the zeros of special functions from a fixed point method a scheme for the computation of the zeros of special functions and orthogonal polynomials is developed we study the structure of the first order difference differential equations ddes satisfied by two fundamental sets of solutions of second order odes y n prime prime x a n x y n x 0 n being the order of the solutions and an x a family of continuous functions it is proved that with a convenient normalization of the solutions t pm1 z z pm mbox rm sign d arctan y n x z y n pm 1 x z are globally convergent iterations with fixed points z x n i x n i being the zeros of yn x d is one of the coefficients in the ddes and z x is a primitive of d the structure of the ddes is also used to set global bounds on the differences between adjacent zeros of functions of consecutive orders and to find iteration steps which guarantee that all the zeros inside a given interval can be found with certainty as an illustration we describe how to implement this scheme for the calculation of the zeros of arbitrary solutions of the bessel coulomb legendre hermite and laguerre equations

active microwave imaging a model based approach a model based approach that improves the spatial resolution of a buried inhomogeneous object using an active microwave imaging system is discussed algorithms are developed for quasi real time processing using a microwave camera specifically constructed for civil engineering applications using reflection mode imaging the model is generated by numerically solving an integral equation representing the ideal scattered field of the object the model is used to develop an optimal spatial filter to process the measured field finally the inverse problem is solved using an imaging algorithm based on diffraction tomography and fourier transforms experimental results indicate a reasonable performance of the approach

uniform upper and lower bounds on the zeros of bessel functions of the first kind we derive upper and lower bounds to the kth positive zero of the bessel function of the first kind of order v these bounds are close to the known asymptotic formulas which are valid only when k or v are very large applications are discussed and upper and lower bounds to the zeros of the airy function are also derived

continuous ranking of zeros of special functions abstract we reexamine and continue the work of j vosmansky j vosmansk zeros of solutions of linear differential equations as continuous functions of the parameter k in j wiener j k hale eds partial differential equations proceedings of conference edinburg tx 1991 in pitman res notes math ser vol 273 1992 pp 253 257 on the concept of continuous ranking of zeros of certain special functions from the point of view of the transformation theory of second order linear differential equations this leads to results on higher monotonicity of such zeros with respect to the rank and to the evaluation of some definite integrals the applications are to airy bessel and hermite functions

application of generalized pade approximants to the special function evaluation problem abstract a new approach to generalized pade approximants is presented its main interest lies in the fact that it avoids instability and or round off errors which occur in traditional algorithms for evaluating pade approximants moreover the approximants are readily obtained applications concerned with the tabulation of some transcendental functions in their critical domain weber and bessel functions illustrate the power of the method

the numerical computation of the confluent hypergeometric functionu a b z an algorithm is given for the computation of the confluent hypergeometric functionu a b z for real values ofa b andz z 0 algol 60 procedures are given the computations are based on a miller algorithm and on asymptotic expansions

computing zeros and orders of bessel functions abstract we consider computing a prescribed number of smallest positive zeros of bessel functions and of their derivatives of a prescribed order within a prescribed relative error we also consider an inverse problem of computing the order of the bessel function that has a zero of a prescribed order at a prescribed positive value the case of bessel functions of real noninteger order less than 1 is also discussed our emphasis in this paper is on algorithm construction and convergence analysis that will be needed for the construction of software for solving the stated problems

error analysis for the computation of zeros of regular coulomb wave function and its first derivative in 1975 one of the coauthors ikebe showed that the problem of computing the zeros of the regular coulomb wave functions and their derivatives may be reformulated as the eigenvalue problem for infinite matrices approximation by truncation is justified but no error estimates are given there the class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by ikebe et al in 1993 where an extremely accurate asymptotic error estimate is shown in this paper we apply this error formula to the former case to obtain error formulas in a closed explicit form

asymptotic relation for zeros of cross product of bessel functions and applications abstract let a k be the k th positive zero of the cross product of bessel functions j r z y z j z y r z where 0 and r 1 we derive an initial value problem for a first order differential equation whose solution x characterizes the limit behavior of a k in the following sense lim k a k x k k x x 0 moreover we show that a k k r 1 2 r we use x to obtain an explicit expression of the pleijel constant for planar annuli and compute some of its values

on a conjecture regarding the extrema of bessel functions and its generalization abstract it was conjectured by a elbert in j comput appl math 133 2001 65 83 that given two consecutive real zeros of a bessel function c of order j and j 1 the zero of the derivative between such two zeros j satisfies j j j 1 we prove that this inequality holds for any bessel function of any real order in addition to these lower bounds upper bounds are obtained in this way we bracket the zeros of the derivative it is discussed how similar relations can be obtained for other special functions which are solutions of a second order ode in particular the case of the zeros of c x c is considered

on the variation with respect to a parameter of zeros of bessel and q bessel functions abstract we apply what is essentially the hellmann feynman theorem in a discrete setting to derive representations for the derivatives with respect to a parameter of the positive zeros of a family of entire functions these representations prove that the zeros are monotone functions of the parameters involved this family includes the bessel functions jv x and their basic analog jv 2 x q as a by product we show that j v 1 2 v 1 increases with v when v 1 jv 1 being the smallest positive zero of jv x we also use the lommel polynomials to derive improved lower bounds for the smallest positive zero of jv x

flat top footprint pattern synthesis through the design of arbitrary planar shaped apertures the problem of generating a flat top main beam with an arbitrary footprint for array elements placed in an arbitrary planar aperture is considered in this paper some simplifying properties of the bessel functions encourages the general framework of the paper to encompass patterns produced by circular aperture and eventually generalize it to arbitrary aperture geometries in this regard two synthesis methods are presented the first method is based on the use of the rayleigh quotient to obtain constant phase array patterns hence a class of generally linear phase patterns can be considered the second approach is based on power pattern synthesis where there is no restriction on the phase of the pattern hence it provides us with greater flexibility the nonlinear problem is appropriately modeled and formulated for amenable performance these two new methods can exhibit a significant reduction in the number of unknown parameters and high flexibility in shaping the desired main beam by arbitrary lattice geometry

locating and computing in parallel all the simple roots of special functions using pvm an algorithm is proposed for locating and computing in parallel and with certainty all the simple roots of any twice continuously di0erentiable function in any speci1c interval to compute with certainty all the roots the proposed method is heavily based on the knowledge of the total number of roots within the given interval to obtain this information we use results from topological degree theory and in particular the kronecker picard approach this theory gives a formula for the computation of the total number of roots of a system of equations within a given region which can be computed in parallel with this tool in hand we construct a parallel procedure for the localization and isolation of all the roots by dividing the given region successively and applying the above formula to these subregions until the 1nal domains contain at the most one root the subregions with no roots are discarded while for the rest a modi1cation of the well known bisection method is employed for the computation of the contained root the new aspect of the present contribution is that the computation of the total number of zeros using the kronecker picard integral as well as the localization and computation of all the roots is performed in parallel using the parallel virtual machine pvm pvm is an integrated set of software tools and libraries that emulates a general purpose 7exible heterogeneous concurrent computing framework on interconnected computers of varied architectures the proposed algorithm has large granularity and low synchronization and is robust it has been implemented and tested and our experience is that it can massively compute with certainty all the roots in a certain interval performance information from massive computations related to a recently proposed conjecture due to elbert this issue j comput appl math 133 2001 65 83 is reported c 2001 elsevier science b v all rights reserved

computing the zeros and turning points of solutions of second order homogeneous linear odes algorithms to compute the zeros and turning points of the solutions of second order odes y prime prime b x y prime a x y 0 are developed two fixed point methods are introduced the first method consists of fixed point iterations that are built from the first order differential system relating the problem function with a contrast function w the contrast function w has zeros interlaced with those of y and it is a solution of a different second order ode this method which generalizes previous findings j segura siam j numer anal 40 2002 pp 114 133 requires the evaluation of the ratio of functions y w the second method is based on fixed point iterations stemming from the second order ode it requires the computation of the logarithmic derivative y prime y both are quadratically convergent methods error bounds are provided the particular case of second order odes depending on one parameter y n prime prime b n x y n prime a n x y n x 0 with applications to the computation of the zeros and turning points of special functions is discussed in detail the combination of both methods provides algorithms for the efficient computation of the zeros and turning points of a broad family of special functions including hypergeometric and confluent hypergeometric functions of real parameters and variables jacobi laguerre and hermite polynomials are particular cases bessel airy coulomb and conical functions among others we provide numerical examples showing the efficiency of the methods

further results on mcmahon s asymptotic approximations we consider the sum of the first two or three terms from the mcmahon asymptotic expansion of the zeros of the cylinder function c x j x cos y x sin 0 and study when this sum represents as an upper or lower bound for the corresponding zero the results established extend in particular the case of the zeros of j x when we recover the inequalities found by forster and petras forster k j and petras k 1993 zamm 73 232 6 for our approach is based on a sturmian comparison theorem discussed in section 2

asymptotics for thenth degree laguerre polynomial evaluated atn we investigate the asymptotic behaviour of n n n where n x denotes the laguerre polynomial of degreen our results give a partial answer to the conjecture n n 1 forn 6 made in 1984 by van iseghem we also show the connection between this conjecture and the continued fraction approximants of 6 sqrt 3 mathord left vphantom 3 pi right kern nulldelimiterspace pi

border collision bifurcations in two dimensional piecewise smooth maps recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps that cannot be classified among the generic cases like saddle node pitchfork or hopf bifurcations occurring in smooth maps in this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision we then present a systematic analysis of such bifurcations by deriving a normal form the piecewise linear approximation in the neighborhood of the border we show that there can be eleven qualitatively different types of border collision bifurcations depending on the parameters of the normal form and these are classified under six cases we present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur this theoretical framework will help in explaining bifurcations in all systems which can be represented by two dimensional piecewise smooth maps

best possible upper bounds for the first two positive zeros of the bessel function j n x the infinite case abstract the first positive zero jv 1 of the bessel function jv x has the asymptotic expansion j v 1 v a 1 2 1 3 v 1 3 3 20 a 2 1 2 1 3 v 1 3 where a1 2 33811 is the first negative zero of the airy function ai x recently lorch has conjectured that the sum of the first three terms in the expansion gives an upper bound for jv 1 i e j v 1 a 1 2 1 3 v 1 3 3 20 a 2 1 2 1 3 v 1 3 and that similar bounds hold for jv k j v k yv k and y v k k 1 2 the objective of this paper is to show that lorch s conjecture is true when v 10 for jv 1 and jv 2

asymptotic numerical approximations for highly oscillatory second order differential equations by the phase function method abstract asymptotic approximations of phase functions for linear second order differential equations whose solutions are highly oscillatory can be obtained using bor vka s theory of linear differential transformations coupled to liouville green wkb asymptotics a numerical method very effective in case of asymptotically polynomial coefficients is extended to other cases of rapidly growing coefficients zeros of solutions can be computed without prior evaluation of the solutions themselves but the method can also be applied to initial and boundary value problems as well as to the case of forced oscillations numerical examples are given to illustrate the performance of the algorithm in all cases the error turns out to be of the order of that made approximating the phase functions

bifurcation analysis of an impact oscillator with a one sided elastic constraint near grazing in this paper a linear oscillator undergoing impact with a secondary elastic support is studied experimentally and semi analytically for near grazing conditions the experimentally observed bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces smooth as well as nonsmooth bifurcations are observed and the resulting atypical bifurcations are explained often as an interplay between them in order to understand the observed bifurcation scenarios a global analysis is required due to the influence of stable and unstable orbits which are born in distant bifurcations but become important at near grazing conditions the good degree of correspondence between experiment and theory fully justifies the modelling approach

bounds on differences of adjacent zeros of bessel functions and iterative relations between consecutive zeros bounds for the distance c s c 1 s between adjacent zeros of cylinder functions are given s and s are such that c s c s c 1 s c k stands for the kth positive zero of the cylinder bessel function c x cos j x sin y x 0 r these bounds together with the application of modified global newton methods based on the monotonic functions f x x2 1c x c 1 x and g x x 2 1 c x c 1 x give rise to forward c k c k 1 and backward c k 1 c k iterative relations between consecutive zeros of cylinder functions the problem of finding all the positive real zeros of bessel functions c x for any real and inside an interval x1 x2 x1 0 is solved in a simple way

convexity of the extreme zeros of gegenbauer and laguerre polynomials let cn x n 0 1 be the ultraspherical gegenbauer polynomials orthogonal in 1 1 with respect to the weight function 1 x2 1 2 denote by xnk k 1 n the zeros of cn x enumerated in decreasing order in this short note we prove that for any n n the product 1 3 2xn1 is a convex function of if 0 the result is applied to obtain some inequalities for the largest zeros of cn x if xnk k 1 n are the zeros of laguerre polynomial ln x also enumerated in decreasing order we prove that xn1 1 is a convex function of for 1

an asymptotic relation for the zeros of bessel functions abstract we define the function j for all real 0 as follows for 1 2 the j denotes the kth positive zero of the bessel function j z of first kind and for k 1 1 l x lim j x and we prove among other things the inequality j moreover we find the first three terms of the asymptotic expansion of x for large values of x and other properties of this function

locating and computing all the simple roots and extrema of a function this paper describes and analyzes two algorithms for locating and computing with certainty all the simple roots of a twice continuously differentiable function f colon a b subset r to r and all the extrema of a three times continuously differentiable function in a b the first algorithm locates and computes all the simple roots or all the extrema while the second one is more efficient in the case where both simple roots and extrema are required this paper also gives analytical estimation of the expected complexity of the algorithms based on the distribution of the roots in a b here only the case of uniform distribution is examined which is also the approach to be followed when no statistical data are available for the function at hand the algorithms have been implemented and tested performance information for a well known bessel function is reported

some recent results on the zeros of bessel functions and orthogonal polynomials several recent results are encountered on the zeros of bessel functions and of the classical orthogonal polynomials many of them are waiting for being published four problems are formulated as conjectures

on estimates for the weights in gaussian quadrature in the ultraspherical case in this paper the christoffel numbers a 2 g for ultraspherical weight functions w a wa x 1 x2 1 2 are investigated using only elementary functions we state new inequalities monotonicity properties and asymptotic approximations which improve several known results in particular denoting by 0 i the trigonometric representation of the gaussian nodes we obtain for a e 0 11 the inequalities n v fn 2 n a 2 sin2 a ag 7r 2a1 3 av n 1 2 a a g 7r 2a a sin 0 in v n n a f n 2 n a 2sin 2 v n a i a 3 a i a 2 4 sin 2 0 a n 8 n a 4 sin 46 o n

efficiently computing many roots of a function we present a new bisection based method for counting and computing roots of a function in a given interval our method is focused on very large problems i e instances with the number of roots of the order of hundreds the method draws its power from the fact that the roots are expected to be many and is able to discover a large percentage of them very efficiently its main advantage apart from its efficiency is the fact that it requires only the sign of the function at a certain point and not its actual value also its simplicity makes it a suitable preprocessing step for reducing the size of the problem prior to more robust but also more demanding methods the algorithm is accompanied by a probabilistic analysis of its behavior which shows that a simple existence criterion like bolzano s rule can be a powerful tool in the zerofinding process

coulomb and bessel functions of complex arguments and order abstract the coulomb wavefunctions originally constructed for real 0 real and integer 0 are defined for and all complex we examine the complex continuation of a variety of series and continued fraction expansions for the coulomb functions and their logarithmic derivatives and then see how these expansions may be selectively combined to calculate both the regular and irregular functions and their derivatives the resulting algorithm 46 is a complex generalisation of steed s method 6 7 as it appears in the real procedure coulfg 10 complex whittaker confluent hypergeometric and bessel functions can also be calculated

monotonicity and concavity properties of zeros of bessel functions abstract let j k y k and c k denote the kth positive zeros of the bessel functions j x y x and of the general cylinder function c x respectively we show among other things that for k 2 3 and 0 c0k and c k v 2 c 0k decreases as increases in the cases of j k and y k these results hold also for k 1

on a special type of border collision bifurcations occurring at infinity abstract in piecewise smooth dynamical systems the regions of existence of a periodic orbit are typically parameter sub spaces confined by border collision bifurcations of this orbit we demonstrate that additionally to the usual border collision bifurcations occurring at finite points in the state space there exist also border collision bifurcations occurring at infinity

singularities in soft impacting systems abstract in this paper the character of the normal form map in the neighbourhood of a grazing orbit is investigated for four possible configurations of soft impacting systems it is shown that if the spring in the impacting surface is relaxed the impacting side of the map has a power of 3 2 but if the spring is pre stressed the map has a square root singularity the singularity appears only in the trace of the jacobian matrix and not in the determinant under all conditions the determinant of the jacobian matrix varies continuously across the grazing condition however if the impacting surface has a damper the determinant decreases exponentially with increasing penetration it is found that the system behaviour is greatly dependent upon a parameter m given by 2 0 forcing and that the singularity disappears for integer values of m thus if the parameters are chosen to obtain an integer value of m one can expect no abrupt change in behaviour as the system passes through the grazing condition from a non impacting mode to an impacting mode with increasing excitation amplitude the above result has been tested on an experimental rig which showed a persistence of a period 1 orbit across the grazing condition for integer values of m but an abrupt transition to a chaotic orbit or a high period orbit for non integer values of m finally through simulation it is shown that the condition for vanishing singularity is not a discrete point in the parameter space this property is valid over a neighbourhood in the parameter space which shrinks for larger values of the stiffness ratio k 2 k 1

on the origin and bifurcations of stick slip oscillations a recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis particular emphasis is put on the bifurcations associated with the appearance of stick

global dynamics of a vibro impacting linear oscillator abstract the steady state vibro impacting responses of one dimensional harmonically excited linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation the steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction unlike the free harmonically excited oscillator the phase space of a vibro impacting system may be inhabited by many attracting sets for example there are sub harmonic multi impact periodic orbits and chaotic steady state responses in order to build a qualitative understanding of vibro impact response an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges use is made of the poincare section or stroboscopic mapping technique essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer the qualitative understanding gained from the analysis and simulations is discussed in an engineering context

compressive sensing imaging of non sparse 2d scatterers by a total variation approach within the born approximation the problem of imaging two dimensional non sparse scatterers with piecewise constant contrast is solved through an innovative total variation compressive sensing tv cs technique under born approximation ba the time harmonic two dimensional inverse scattering problem is reformulated into an equivalent augmented lagrangian one according to the tv cs formulation the resulting unconstrained functional is then minimized by means of a deterministic alternating direction algorithm selected results from a numerical assessment are discussed to give the interested reader some insights on the accuracy the flexibility the robustness and the efficiency of the tv cs inversion scheme also in comparison with available implementations of state of the art inversion methods still formulated within the ba

pade type approximants of exp z whose denominators are 1 z n n in this paper we show that the sequences of pade type approximants k 1 k and k k converge to exp z uniformly and geometrically on every compact subset of the plane a numerical study has been done which discriminates these sequences from the point of view ofa acceptability

higher order zero time discontinuity mapping for analysis of degenerate grazing bifurcations of impacting oscillators abstract the discontinuity mapping technique provides quite a good way to study the near grazing dynamics in the impacting oscillators however disagreements between the results obtained by the discontinuity mapping technique and the direct numerical observations still exist in terms of modeling and stability analysis in particular when applying the existing theory to deal with the complicated degenerate or other co dimension two grazing bifurcations even potential mistake in the qualitative analysis might arise to reduce such disagreements the local zero time discontinuity mapping is extended to higher order terms for the generic multi degree of freedom impacting oscillators under harmonic excitation in this paper furthermore to demonstrate the necessity and feasibility of using the higher order zero time discontinuity mapping for the analysis of degenerate grazing bifurcation comparisons between the lower and higher order truncation are made through the existence analysis of near grazing impact periodic motions

inequalities and monotonicity properties for zeros of hermite functions we study the variation of the zeros of the hermite function h t with respect to the positive real variable we show that for each nonnegative integer n h t has exactly n 1 real zeros when n n 1 and that each zero increases from i1 to 1 as increases we establish a formula for the derivative of a zero with respect to the parameter this derivative is a completely monotonic function of by products include some results on the regular sign behaviour of difierences of zeros of hermite polynomials as well as a proof of some inequalities related to work of w k hayman and e l ortiz for the largest zero of h t similar results on zeros of certain con uent hypergeometric functions are given too these specialize to results on the flrst second etc positive zeros of hermite polynomials

an approximation for the zeros of bessel functions letc vk be thekth positive zero of the cylinder functionc v x cos j v x sin y v x wherej v x y v x are the bessel functions of first kind and second kind resp andv 0 0 definej vk byj vk c vk with kappa k frac alpha pi using the notation 1 k we derive the first two terms of the asymptotic expansion ofj vk in terms of the powers of at the expense of solving a transcendental equation numerical examples are given to show the accuracy of this approximation

a computational study of reconstruction algorithms for diffraction tomography interpolation versus filtered backpropagation from the standpoint of reporting a new contribution this paper shows that by using bilinear interpolation followed by direct two dimensional fourier inversion one can obtain reconstructions of quality which is comparable to that produced by the filtered backpropagation algorithm proposed recently by devaney for an n n image reconstructed from n diffracted projections the former approach requires approximately 4n fft s whereas the backpropagation technique requires approximately n2fft s we have also taken this opportunity to present the reader with a tutorial introduction to diffraction tomography an area that is becoming increasingly important not only in medical imaging but also in underwater and seismic mapping with microwaves and sound the main feature of the tutorial part is the statement of the fourier diffraction projection theorem which is an extension of the traditional fourier slice theorem to the case of image formation with diffracting illumination

a singular sturm liouville equation under homogeneous boundary conditions abstract given 0 and f l 2 0 1 we are interested in the equation x 2 u x u x f x on 0 1 u 1 0 we prescribe appropriate weighted homogeneous boundary conditions at the origin and prove the existence and uniqueness of h loc 2 0 1 solutions we study the regularity at the origin of such solutions we perform a spectral analysis of the differential operator l u x 2 u u under those appropriate homogeneous boundary conditions

a differential inequality for the positive zeros of bessel functions abstract let jv k be the kth positive zero of the bessel function jv z of the first kind and order v 1 we prove the differential inequalities where hn v x n 0 are the lommel polynomials watson 1966 as a consequence we obtain some new lower and upper bounds for jv k improving previously known results

on the zeros and turning points of special functions globally convergent fixed point iterations together with bounds on differences of zeros from sturm methods are used to build efficient algorithms for the computation of the zeros of special functions satisfying first order linear difference differential equations bounds on the spacing between the zeros are obtained as a by product turning points can also be computed in a similar way new analytical information is also obtained in this case which for instance can be used to prove a conjecture by elbert on the turning points of bessel functions

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

zeros of gegenbauer and hermite polynomials and connection coefficients in this paper sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of obrechkoff and certain explicit connection coefficients for these polynomials as a consequence sharp bounds for the zeros of the hermite polynomials are obtained

automatic computation of zeros of bessel functions and other special functions an efficient algorithm is developed for the automatic calculation of the zeros of bessel functions of arbitrary order coulomb wave functions and other special functions remarkably this can be done without any actual functional evaluations limitations and generalizations are discussed

zeros of classical orthogonal polynomials of a discrete variable in this paper we obtain sharp bounds for the zeros of classical orthogonal polynomials of a discrete variable considered as functions of a parameter by using a theorem of a markov and the so called hellmannfeynman theorem comparisons with previous results for zeros of hahn meixner kravchuk and charlier polynomials are also presented

intrinsic cylindrical and spherical waves intrinsic waveforms associated with cylindrical and spherical bessel functions are obtained by eliminating the factors responsible for the inverse radius and inverse square radius laws of wave power per unit area of wavefront the resulting expressions are riccati bessel functions for both cases and these can be written in terms of amplitude and phase functions of order v and wave variable z when z is real it is shown that a spatial phase angle of the intrinsic wave can be defined and this together with its amplitude function is systematically investigated for a range of fixed orders and varying z the derivatives of riccati bessel functions are also examined all the component functions exhibit different behaviour in the near field depending on the order being less than equal to or greater than 1 2 plots of the phase angle can be used to display the locations of the zeros of the general riccati bessel functions and lead to new relations concerning the ordering of the real zeros of bessel functions and the occurrence of multiple zeros when the argument of the bessel function is fixed

reliable computation of the zeros of solutions of second order linear odes using a fourth order method a fourth order fixed point method to compute the zeros of solutions of second order homogeneous linear odes is obtained from the approximate integration of the riccati equation associated with the ode the method requires the evaluation of the logarithmic derivative of the function and also uses the coefficients of the ode an algorithm to compute with certainty all the zeros in an interval is given which provides a fast reliable and accurate method of computation the method is illustrated by the computation of the zeros of gauss hypergeometric functions including jacobi polynomials and confluent hypergeometric functions laguerre polynomials hermite polynomials and bessel functions included among others the examples show that typically 4 or 5 iterations per root are enough to provide more than 100 digits of accuracy without requiring a priori estimations of the roots

a connection between ultraspherical and pseudo ultraspherical polynomials abstract the pseudo ultraspherical polynomial of degree n is defined by c n x i n c n i x where c n x is the ultraspherical polynomial we derive a simple expression linking the polynomials c n and c n 1 2 n and show how to derive various properties of the zeros of c n when 1 n from properties of the zeros of c n when 1 2