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#REDIRECT [[Hyperbolic functions]]
In mathematics, the '''tanhc function''' is defined as<ref>Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TanhcFunction.html</ref>
{{Rcat shell|
: <math>\operatorname{tanhc}(z)=\frac {\tanh(z) }{z}</math>
{{R to related topic}}
[[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]]
;Imaginary part in complex plane
*<math> \operatorname{Im} \left( \frac {\tanh(x+iy) }{x+iy} \right) </math>
;Real part in complex plane
*<math> \operatorname{Re} \left( \frac {\tanh \left( x+iy \right) }{x+iy} \right) </math>
;absolute magnitude
*<math> \left| \frac {\tanh(x+iy) }{x+iy} \right| </math>
;First-order derivative
*<math> \frac {1- \tanh(z))^2}{z} - \frac {\tanh(z)}{z^2} </math>
;Real part of derivative
*<math> -\operatorname{Re} \left( -\frac {1- (\tanh(x+iy))^2}{x+iy} +\frac{\tanh(x+iy)}{(x+iy)^2} \right)
</math>
;Imaginary part of derivative
*<math>-\operatorname{Im} \left( -\frac {1-(\tanh(x+iy))^2}{x+iy} + \frac {\tanh(x+iy)}{(x+iy)^2} \right)
</math>
;absolute value of derivative
*<math> \left| -\frac{1-(\tanh(x+iy))^2}{x+iy}+\frac {\tanh(x+iy)}{(x+iy)^2} \right| </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>
*<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>
* <math>\operatorname{tanhc}(z)= \frac{i{\rm \ WhittakerM}(0,\,1/2,\,2\,z)}{{\rm WhittakerM}(0,\,1/2,\,i\pi -2\,z)} z</math>
==Series expansion==
: <math>\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>
:<math>\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>
==Pade approximation==
<math> {\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}
</math>
==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]]
|
|}
{|
|[[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]]
[[Tanhc integral]]
[[Coshc function]]
==References==
<references/>
[[Category:Special functions]]
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