Coshc函数

Coshc函数常见于有关光学散射、海森堡时空和双曲几何学的论文中其定义如下： Coshc ⁡ ( z ) = cosh ⁡ ( z ) z {\displaystyle \operatorname {Coshc} (z)={\frac {\cosh(z)}{z}}} 它是下列微分方程的一个解： w ( z ) z − 2 d d z w ( z ) − z d 2 d z 2 w ( z ) = 0 {\displaystyle w\left(z\right)z-2\,{\frac {d}{dz}}w\left(z\right)-z{\frac {d^{2}}{d{z}^{2}}}w\left(z\right)=0} 复域虚部 Im ⁡ ( cosh ⁡ ( x + i y ) x + i y ) {\displaystyle \operatorname {Im} \left({\frac {\cosh(x+iy)}{x+iy}}\right)} 复域实部 Re ⁡ ( cosh ⁡ ( x + i y ) x + i y ) {\displaystyle \operatorname {Re} \left({\frac {\cosh \left(x+iy\right)}{x+iy}}\right)} 绝对值 | cosh ⁡ ( x + i y ) x + i y | {\displaystyle \left|{\frac {\cosh(x+iy)}{x+iy}}\right|} 一阶导数 sinh ⁡ ( z ) z − cosh ⁡ ( z ) z 2 {\displaystyle {\frac {\sinh(z)}{z}}-{\frac {\cosh(z)}{z^{2}}}} 导数实部 − Re ⁡ ( − 1 − ( cosh ⁡ ( x + i y ) ) 2 x + i y + cosh ⁡ ( x + i y ) ( x + i y ) 2 ) {\displaystyle -\operatorname {Re} \left(-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right)} 导数虚部 − Im ⁡ ( − 1 − ( cosh ⁡ ( x + i y ) ) 2 x + i y + cosh ⁡ ( x + i y ) ( x + i y ) 2 ) {\displaystyle -\operatorname {Im} \left(-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right)} 导数绝对值 | − 1 − ( cosh ⁡ ( x + i y ) ) 2 x + i y + cosh ⁡ ( x + i y ) ( x + i y ) 2 | {\displaystyle \left|-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right|} Coshc ⁡ ( z ) = ( i z + 1 / 2 π ) M ( 1 , 2 , i π − 2 z ) e 1 / 2 i π − z z {\displaystyle \operatorname {Coshc} (z)={\frac {\left(iz+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,i\pi -2\,z\right)}}{{{\rm {e}}^{1/2\,i\pi -z}}z}}} Coshc ⁡ ( z ) = 1 2 ( 2 i z + π ) H e u n B ( 2 , 0 , 0 , 0 , 2 1 / 2 i π − z ) e 1 / 2 i π − z z {\displaystyle \operatorname {Coshc} (z)={\frac {1}{2}}\,{\frac {\left(2\,iz+\pi \right){\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right)}{{{\rm {e}}^{1/2\,i\pi -z}}z}}} Coshc ⁡ ( z ) = − i ( 2 i z + π ) W h i t t a k e r M ( 0 , 1 / 2 , i π − 2 z ) ( 4 i z + 2 π ) z {\displaystyle \operatorname {Coshc} (z)={\frac {-i\left(2\,iz+\pi \right){{\rm {\mathbf {W} hittakerM}}\left(0,\,1/2,\,i\pi -2\,z\right)}}{\left(4\,iz+2\,\pi \right)z}}} Coshc ⁡ z ≈ ( z − 1 + 1 2 z + 1 24 z 3 + 1 720 z 5 + 1 40320 z 7 + 1 3628800 z 9 + 1 479001600 z 11 + 1 87178291200 z 13 + O ( z 15 ) ) {\displaystyle \operatorname {Coshc} z\approx ({z}^{-1}+{\frac {1}{2}}z+{\frac {1}{24}}{z}^{3}+{\frac {1}{720}}{z}^{5}+{\frac {1}{40320}}{z}^{7}+{\frac {1}{3628800}}{z}^{9}+{\frac {1}{479001600}}{z}^{11}+{\frac {1}{87178291200}}{z}^{13}+O\left({z}^{15}\right))} Tanc函数 Tanhc函数 Sinhc函数 PN Den Outer, TM Nieuwenhuizen, A Lagendijk,Location of objects in multiple-scattering media,JOSA A, Vol. 10, Issue 6, pp. 1209-1218 (1993) T Körpinar ,New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime - International Journal of Theoretical Physics, 2014 - Springer Nilg¨un S¨onmez,A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry,International Mathematical Forum, 4, 2009, no. 38, 1877 - 1881 JHM ten Thije Boonkkamp, J van Dijk, L Liu,Extension of the complete flux scheme to systems of conservation laws,J Sci Comput (2012) 53:552–568,DOI 10.1007/s10915-012-9588-5 Weisstein, Eric W. "Coshc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CoshcFunction.html[永久失效連結]