Coshc函数

Coshc函数常见于有关光学散射^[1]、海森堡时空^[2]和双曲几何学的论文中^[3]其定义如下：^[4][5]

${\displaystyle \operatorname {Coshc} (z)={\frac {\cosh(z)}{z}}}$

它是下列微分方程的一个解：

${\displaystyle w\left(z\right)z-2\,{\frac {d}{dz}}w\left(z\right)-z{\frac {d^{2}}{d{z}^{2}}}w\left(z\right)=0}$

Coshc 2D plot

Coshc'(z) 2D plot

复域虚部

• ${\displaystyle \operatorname {Im} \left({\frac {\cosh(x+iy)}{x+iy}}\right)}$

复域实部

• ${\displaystyle \operatorname {Re} \left({\frac {\cosh \left(x+iy\right)}{x+iy}}\right)}$

绝对值

• ${\displaystyle \left|{\frac {\cosh(x+iy)}{x+iy}}\right|}$

一阶导数

• ${\displaystyle {\frac {\sinh(z)}{z}}-{\frac {\cosh(z)}{z^{2}}}}$

导数实部

• ${\displaystyle -\operatorname {Re} \left(-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right)}$

导数虚部

• ${\displaystyle -\operatorname {Im} \left(-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right)}$

导数绝对值

• ${\displaystyle \left|-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right|}$

表示为其他特殊函数

• ${\displaystyle \operatorname {Coshc} (z)={\frac {\left(iz+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,i\pi -2\,z\right)}}{{{\rm {e}}^{1/2\,i\pi -z}}z}}}$

• ${\displaystyle \operatorname {Coshc} (z)={\frac {1}{2}}\,{\frac {\left(2\,iz+\pi \right){\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right)}{{{\rm {e}}^{1/2\,i\pi -z}}z}}}$

• ${\displaystyle \operatorname {Coshc} (z)={\frac {-i\left(2\,iz+\pi \right){{\rm {\mathbf {W} hittakerM}}\left(0,\,1/2,\,i\pi -2\,z\right)}}{\left(4\,iz+2\,\pi \right)z}}}$

级数展开

${\displaystyle \operatorname {Coshc} z\approx ({z}^{-1}+{\frac {1}{2}}z+{\frac {1}{24}}{z}^{3}+{\frac {1}{720}}{z}^{5}+{\frac {1}{40320}}{z}^{7}+{\frac {1}{3628800}}{z}^{9}+{\frac {1}{479001600}}{z}^{11}+{\frac {1}{87178291200}}{z}^{13}+O\left({z}^{15}\right))}$

图集

Coshc abs complex 3D

Coshc Im complex 3D plot

Coshc Re complex 3D plot

Coshc'(z) Im complex 3D plot

Coshc'(z) Re complex 3D plot

Coshc'(z) abs complex 3D plot

Coshc'(x) abs density plot

Coshc'(x) Im density plot

Coshc'(x) Re density plot

参见

• Tanc函数
• Tanhc函数
• Sinhc函数

参考文献

1. PN Den Outer, TM Nieuwenhuizen, A Lagendijk,Location of objects in multiple-scattering media,JOSA A, Vol. 10, Issue 6, pp. 1209-1218 (1993)
2. T Körpinar ,New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime - International Journal of Theoretical Physics, 2014 - Springer
3. Nilg¨un S¨onmez,A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry,International Mathematical Forum, 4, 2009, no. 38, 1877 - 1881
4. 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
5. Weisstein, Eric W. "Coshc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CoshcFunction.html^{[永久失效链接]}