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

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