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：

<math>\operatorname{Coshc}(z)=\frac {\cosh(z) }{z}</math>

它是下列微分方程的一个解：

<math>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</math>

File:Coshc 2D plot.png

Coshc 2D plot

File:Coshc'(z) 2D plot.png

Coshc'(z) 2D plot

复域虚部

• <math> \operatorname{Im} \left( \frac {\cosh(x+iy) }{x+iy} \right) </math>

复域实部

• <math> \operatorname{Re} \left( \frac {\cosh \left( x+iy \right) }{x+iy} \right) </math>

绝对值

• <math> \left| \frac {\cosh(x+iy) }{x+iy} \right| </math>

一阶导数

• <math> \frac {\sinh(z)}{z} - \frac {\cosh(z)}{z^2} </math>

导数实部

• <math> -\operatorname{Re} \left( -\frac {1- (\cosh(x+iy))^2}{x+iy} +\frac{\cosh(x+iy)}{(x+iy)^2} \right)

   </math>

导数虚部

• <math>-\operatorname{Im} \left( -\frac {1-(\cosh(x+iy))^2}{x+iy} + \frac {\cosh(x+iy)}{(x+iy)^2} \right)

    </math>

导数绝对值

• <math> \left| -\frac{1-(\cosh(x+iy))^2}{x+iy}+\frac {\cosh(x+iy)}{(x+iy)^2} \right| </math>

表示为其他特殊函数

• <math>\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}} </math>

• <math>\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}}

</math>

• <math>\operatorname{Coshc}(z)= {\frac {-i \left( 2\,iz+\pi \right)

{{\rm \mathbf WhittakerM}\left(0,\,1/2,\,i\pi -2\,z\right)}}{ \left( 4\,iz+2\,\pi

\right) z}}

</math>

级数展开

<math>\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) )</math>

图集

File:Coshc abs complex 3D plot.png Coshc abs complex 3D

File:Coshc Im complex 3D plot.png Coshc Im complex 3D plot

File:Coshc Re complex 3D plot.png Coshc Re complex 3D plot

File:Coshc'(z) Im complex 3D plot.png Coshc'(z) Im complex 3D plot

File:Coshc'(z) Re complex 3D plot.png Coshc'(z) Re complex 3D plot

File:Coshc'(z) abs complex 3D plot.png Coshc'(z) abs complex 3D plot

File:Coshc'(x) abs density plot.JPG Coshc'(x) abs density plot

File:Coshc'(x) Im density plot.JPG Coshc'(x) Im density plot

File:Coshc'(x) Re density plot.JPG Coshc'(x) Re density plot

参见

• Tanc函数
• Tanhc函数
• Sinhc函数