Tanhc function

The cardinal hyperbolic tangent function tanhc(z) plotted in the complex plane from -2-2i to 2+2i

In mathematics, the tanhc function is defined for [math]\displaystyle{ z \neq 0 }[/math] as^[1] [math]\displaystyle{ \operatorname{tanhc}(z)=\frac {\tanh(z) }{z} }[/math]The tanhc function is the hyperbolic analogue of the tanc function.

Tanhc 2D plot

Tanhc'(z) 2D plot

Tanhc integral 2D plot

Tanhc integral 3D plot

Properties

The first-order derivative is given by

[math]\displaystyle{ \frac {\operatorname{sech}^2(z)}{z} - \frac {\tanh(z)}{z^2} }[/math]

The Taylor series expansion[math]\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) }[/math]which leads to the series expansion of the integral as[math]\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) ) }[/math]

The Padé approximant is[math]\displaystyle{ \operatorname{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} }[/math]

In terms of other special functions

[math]\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} }} }[/math], where [math]\displaystyle{ {\rm{KummerM}}(a,b,z) }[/math] is Kummer's confluent hypergeometric function.
[math]\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}} }[/math], where [math]\displaystyle{ {\rm{HeunB}}(q, \alpha, \gamma, \delta, \epsilon ,z) }[/math] is the biconfluent Heun function.
[math]\displaystyle{ \operatorname{tanhc}(z)= \frac{i{\rm \ WhittakerM}(0,\,1/2,\,2\,z)}{{\rm WhittakerM}(0,\,1/2,\,i\pi -2\,z)} z }[/math], where [math]\displaystyle{ {\rm{WhittakerM}}(a,b,z) }[/math] is a Whittaker function.