Sinhc函数

数学上的Sinhc函数定义如下^[1][2][3]

${\displaystyle \operatorname {Sinhc} (z)={\frac {\sinh(z)}{z}}}$，它是下列微分方程的一个解：

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

Sinhc 2D plot

Sinhc'(z) 2D plot

Sinhc integral 2D plot

复域虚部

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

复域实部

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

绝对值

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

一阶导数

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

导数实部

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

导数虚部

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

绝对值

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