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In mathematics, the '''Tanhctanhc function''' is defined as<ref>Weisstein, Eric W. "Tanhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TanhcFunction.html </ref>
: <math>\operatorname{tanhc}(z)=\frac {\tanh \left( z \right) }{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]]
*<math> tanhc(z)={\frac {\tanh \left( z \right) }{z}} </math>
;Imaginary part in complex plane
*<math> \operatorname{\it Im} \left( {\frac {\tanh \left( x+iy \right) }{x+iy}} \right) </math>
;Real part in complex plane
*<math>{\it \operatorname{Re} \left( {\frac {\tanh \left( x+iy \right) }{x+iy}} \right) </math>
;absolute magnitude
*<math> \left| {\frac {\tanh \left( x+iy \right) }{x+iy}} \right| </math>
;First -order derivative
*<math>{ \frac {1- \left( \tanh \left( z \right) \right) ^{2}}{z}} -{ \frac {\tanh(z)}{z^2} </math>
\tanh \left( z \right) }{{z}^{2}}}
</math>
;Real part of derivative
*<math> -\operatorname{\it Re} \left( -{\frac {1- \left( \tanh(x+iy))^2}{x+iy} +\leftfrac{\tanh( x+iy)}{(x+iy)^2} \right)
\right) ^{2}}{x+iy}}+{\frac {\tanh \left( x+iy \right) }{ \left( x+iy
\right) ^{2}}} \right)
</math>
;Imaginary part of derivative
*<math>-\operatorname{\it Im} \left( -{\frac {1- (\lefttanh(x+iy))^2}{x+iy} + \tanhfrac {\lefttanh( x+iy)}{(x+iy)^2} \right)
\right) ^{2}}{x+iy}}+{\frac {\tanh \left( x+iy \right) }{ \left( x+iy
\right) ^{2}}} \right)
</math>
;absolute value of derivative
*<math> \left| -{\frac {1- \left( \tanh \left( x+iy \right) \right) ^{2}{x+iy}+\frac {\tanh(x+iy)}{(x+iy)^2} \right| </math>
iy}}+{\frac {\tanh \left( x+iy \right) }{ \left( x+iy \right) ^{2}}}
\right|
</math>
==In terms of other special functions==
*<math>tanhc(z)=2\,{\frac {{{\rm KummerM}\left(1,\,2,\,2\,z\right)}}{ \left( 2\,iz+\pi
* <math>\rightoperatorname{tanhc}(z)=2\,{\frac {{{\rm KummerM}\left(1,\,2,\,i\pi -2\,z\right)}}{(2\,iz+\pi) {\rm eKummerM}(1,\,2,\,i\pi -2\,z) e^{2\,z-1/2\,i\pi} }}}}</math>
*<math>tanhc(z)=2\,{\frac {{\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {z} \right) }{
*<math>\leftoperatorname{tanhc}( z)=2 \frac {\operatorname{HeunB}(2,0,0,0,iz+\pi sqrt{2} \rightsqrt{z}) }{( 2iz+\itpi) \operatorname{HeunB} \left( 2,0,0,0,\sqrt {2} \sqrt{1/2\,i\pi -z}) e^{2\,z-1/2\,i\pi}} </math>
1/2\,i\pi -z} \right) {{\rm e}^{2\,z-1/2\,i\pi }}}}</math>
* <math>\operatorname{tanhc}(z)={ \frac {i{{\rm \ WhittakerM}\left(0,\,1/2,\,2\,z\right)}}{{\rm WhittakerM}(0,\,1/2,\,i\pi -2\,z)} z</math>
{{\rm WhittakerM}\left(0,\,1/2,\,i\pi -2\,z\right)}z}}</math>
==Series expansion==
<math>tanhc \approx (1-{\frac {1}{3}}{z}^{2}+{\frac {2}{15}}{z}^{4}-{\frac {17}{315}}{z}^{
6: <math>\operatorname{tanhc}+ z \approx \left(1-\frac{1}{3} z^2 + \frac {622}{283515} z^4 - \frac {17}{z315} z^6 + \frac {862}-{2835} z^8 - \frac {1382}{155925}}{ z}^{10} + \frac {21844}{6081075} z^{12} - \frac {929569}{638512875} z^{14}+O(z^{16}) \right)</math>
21844}{6081075}}{z}^{12}-{\frac {929569}{638512875}}{z}^{14}+O \left(
{z}^{16} \right) )</math>
==Gallery==
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