Tanhc函數

Tanhc函數定義如下^[1]

Tanhc 2D plot

Tanhc'(z) 2D

Tanhc 積分圖

Tanhc integral 3D plot

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

復域虛部

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

復域實部

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

復域絕對值

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

一階微商

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

微商實部

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

微商虛部

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

微商絕對值

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

積分函數

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

用其他特殊函數表示

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

級數展開

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

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

圖像

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

參看

• Sinhc 函數
• Coshc 函數
• Tanc 函數
• 雙曲正弦積分函數