Tanhc function

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

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

Imaginary part in complex plane

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

Real part in complex plane

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

absolute magnitude

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

First-order derivative

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

Real part of derivative

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

Imaginary part of derivative

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

absolute value of derivative

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

In terms of other special functions

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

Series expansion

\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)

\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

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