Tanhc函数

Tanhc函数定义如下^[1]

Tanhc 2D plot

Tanhc'(z) 2D

Tanhc 积分图

Tanhc integral 3D plot

$tanhc(z)={\frac {\tanh \left(z\right)}{z}}$

复域虚部

${\it {Im}}\left({\frac {\tanh \left(x+iy\right)}{x+iy}}\right)$

复域实部

${\it {Re}}\left({\frac {\tanh \left(x+iy\right)}{x+iy}}\right)$

复域绝对值

$\left|{\frac {\tanh \left(x+iy\right)}{x+iy}}\right|$

一阶微商

${\frac {1-\left(\tanh \left(z\right)\right)^{2}}{z}}-{\frac {\tanh \left(z\right)}{{z}^{2}}}$

微商实部

$-{\it {Re}}\left(-{\frac {1-\left(\tanh \left(x+iy\right)\right)^{2}}{x+iy}}+{\frac {\tanh \left(x+iy\right)}{\left(x+iy\right)^{2}}}\right)$

微商虚部

$-{\it {Im}}\left(-{\frac {1-\left(\tanh \left(x+iy\right)\right)^{2}}{x+iy}}+{\frac {\tanh \left(x+iy\right)}{\left(x+iy\right)^{2}}}\right)$

微商绝对值

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

积分函数

$\int _{0}^{z}\!{\frac {\tanh \left(x\right)}{x}}{dx}$

用其他特殊函数表示

$tanhc(z)=2\,{\frac {{\rm {KummerM}}\left(1,\,2,\,2\,z\right)}{\left(2\,iz+\pi \right){{\rm {KummerM}}\left(1,\,2,\,i\pi -2\,z\right)}{{\rm {e}}^{2\,z-1/2\,i\pi }}}}$
$tanhc(z)=2\,{\frac {{\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {z}}\right)}{\left(2\,iz+\pi \right){\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right){{\rm {e}}^{2\,z-1/2\,i\pi }}}}$
$tanhc(z)={\frac {i{{\rm {\ WhittakerM}}\left(0,\,1/2,\,2\,z\right)}}{{{\rm {WhittakerM}}\left(0,\,1/2,\,i\pi -2\,z\right)}z}}$
$tanhc(z)={\frac {i\left({{\rm {e}}^{2\,z}}-1\right)}{\left({{\rm {e}}^{i\pi -2\,z}}-1\right){{\rm {e}}^{2\,z-1/2\,i\pi }}z}}$

级数展开

$tanhc\approx (1-{\frac {1}{3}}{z}^{2}+{\frac {2}{15}}{z}^{4}-{\frac {17}{315}}{z}^{6}+{\frac {62}{2835}}{z}^{8}-{\frac {1382}{155925}}{z}^{10}+{\frac {21844}{6081075}}{z}^{12}-{\frac {929569}{638512875}}{z}^{14}+O\left({z}^{16}\right))$

$\int _{0}^{z}\!{\frac {\tanh \left(x\right)}{x}}{dx}=(z-{\frac {1}{9}}{z}^{3}+{\frac {2}{75}}{z}^{5}-{\frac {17}{2205}}{z}^{7}+{\frac {62}{25515}}{z}^{9}-{\frac {1382}{1715175}}{z}^{11}+O\left({z}^{13}\right))$

图像

Tanhc abs complex 3D

Tanhc Im complex 3D plot

Tanhc Re complex 3D plot

Tanhc'(z) Im complex 3D plot

Tanhc'(z) Re complex 3D plot

Tanhc'(z) abs complex 3D plot

Tanhc abs plot

Tanhc Im plot

Tanhc Re plot

Tanhc'(z) Im plot

Tanhc'(z) abs plot

Tanhc'(z) Re plot

Tanhc integral abs 3D plot

Tanhc integral Im 3D plot

Tanhc integral Re 3D plot

Tanhc integral abs density plot

Tanhc integral Im density plot

Tanhc integral Re density plot

参看

Sinhc 函数
Coshc 函数
Tanc 函数
双曲正弦积分函数

参考资料

^ Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TanhcFunction.html （页面存档备份，存于互联网档案馆）

[1] Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TanhcFunction.html （页面存档备份，存于互联网档案馆）

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