Tanhc function

In mathematics, the tanhc function is defined as[1]

${\displaystyle \operatorname {tanhc} (z)={\frac {\tanh(z)}{z}}}$

Tanhc 2D plot

Tanhc'(z) 2D plot

Tanhc integral 2D plot

Tanhc integral 3D plot

Imaginary part in complex plane

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

Real part in complex plane

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

absolute magnitude

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

First-order derivative

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

Real part of derivative

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

Imaginary part of derivative

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

absolute value of derivative

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

In terms of other special functions

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

Series expansion

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

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

Padé approximation

${\displaystyle {\it {Tanhc}}\left(z\right)=\left(1+{\frac {7}{51}}\,{z}^{2}+{\frac {1}{255}}\,{z}^{4}+{\frac {2}{69615}}\,{z}^{6}+{\frac {1}{34459425}}\,{z}^{8}\right)\left(1+{\frac {8}{17}}\,{z}^{2}+{\frac {7}{255}}\,{z}^{4}+{\frac {4}{9945}}\,{z}^{6}+{\frac {1}{765765}}\,{z}^{8}\right)^{-1}}$

Gallery

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

See also

• Sinhc function
• Tanc function
• Coshc function

References

1. Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TanhcFunction.html