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#REDIRECT [[Hyperbolic functions]]
[[File:The cardinal hyperbolic tangent function tanhc(z) plotted in the complex plane from -2-2i to 2+2i.svg|alt=The cardinal hyperbolic tangent function tanhc(z) plotted in the complex plane from -2-2i to 2+2i|thumb|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>z \neq 0</math> as<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Tanhc Function |url=https://mathworld.wolfram.com/TanhcFunction.html |access-date=2022-11-17 |website=mathworld.wolfram.com |language=en}}</ref>
<math display="block">\operatorname{tanhc}(z)=\frac {\tanh(z) }{z}</math>
[[File:Tanhc 2D plot.png|thumb|Tanhc 2D plot]]
[[File:Tanhc'(z) 2D plot.png|thumb|Tanhc'(z) 2D plot]]
[[File:Tanhc integral 2D plot.png|thumb|Tanhc integral 2D plot]]
[[File:Tanhc integral 3D plot.png|thumb|Tanhc integral 3D plot]]
{{Rcat shell|
== Properties ==
{{R to related topic}}
The first-order derivative is given by
:<math> \frac {\operatorname{sech}^2(z)}{z} - \frac {\tanh(z)}{z^2} </math>
The [[Taylor series]] expansion<math display="block">\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 display="block">\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 display="block"> \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>\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>{\rm{KummerM}}(a,b,z)</math> is Kummer's [[confluent hypergeometric function]].
*<math>\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>{\rm{HeunB}}(q, \alpha, \gamma, \delta, \epsilon ,z)</math> is the biconfluent [[Heun function]].
* <math>\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>{\rm{WhittakerM}}(a,b,z)</math> is a [[Whittaker function]].
==Gallery==
{|
|[[File:Tanhc abs complex 3D plot.png|thumb|Tanhc abs complex 3D]]
|[[File:Tanhc Im complex 3D plot.png|thumb|Tanhc Im complex 3D plot]]
|[[File:Tanhc Re complex 3D plot.png|thumb|Tanhc Re complex 3D plot]]
{|
|[[File:Tanhc'(z) Im complex 3D plot.png|thumb|Tanhc'(z) Im complex 3D plot]]
|[[File:Tanhc'(z) Re complex 3D plot.png|thumb|Tanhc'(z) Re complex 3D plot]]
|[[File:Tanhc'(z) abs complex 3D plot.png|thumb|Tanhc'(z) abs complex 3D plot]]
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|}
{|
|[[File:Tanhc abs plot.JPG|thumb|Tanhc abs plot]]
|[[File:Tanhc Im plot.JPG|thumb|Tanhc Im plot]]
|[[File:Tanhc Re plot.JPG|thumb|Tanhc Re plot]]
|}
{|
|[[File:Tanhc'(z) Im plot.JPG|thumb|Tanhc'(z) Im plot]]
|[[File:Tanhc'(z) abs plot.JPG|thumb|Tanhc'(z) abs plot]]
|[[File:Tanhc'(z) Re plot.JPG|thumb|Tanhc'(z) Re plot]]
|}
{|
|[[File:Tanhc integral abs 3D plot.png|thumb|Tanhc integral abs 3D plot]]
|[[File:Tanhc integral Im 3D plot.png|thumb|Tanhc integral Im 3D plot]]
|[[File:Tanhc integral Re 3D plot.png|thumb|Tanhc integral Re 3D plot]]
|}
{|
|[[File:Tanhc integral abs density plot.JPG|thumb|Tanhc integral abs density plot]]
|[[File:Tanhc integral Im density plot.JPG|thumb|Tanhc integral Im density plot]]
|[[File:Tanhc integral Re density plot.JPG|thumb|Tanhc integral Re density plot]]
|}
==See also==
* [[Sinhc function]]
* [[Tanc function]]
* [[Coshc function]]
==References==
<references/>
[[Category:Special functions]]
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