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

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

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

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

See also

Sinhc function Tanc function Tanhc function Coshc function

References

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