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