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#REDIRECT [[Hyperbolic functions]]
'''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>tanhc(z)=\frac {\tanh \left( z \right) }{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]]
*<math> tanhc(z)={\frac {\tanh \left( z \right) }{z}} </math>
;Imaginary part in complex plane
*<math> {\it Im} \left( {\frac {\tanh \left( x+iy \right) }{x+iy}} \right) </math>
;Real part in complex plane
*<math>{\it 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 \left( z \right) }{{z}^{2}}}
</math>
;Real part of derivative
*<math> -{\it Re} \left( -{\frac {1- \left( \tanh \left( x+iy \right)
\right) ^{2}}{x+iy}}+{\frac {\tanh \left( x+iy \right) }{ \left( x+iy
\right) ^{2}}} \right)
</math>
;Imaginary part of derivative
*<math>-{\it Im} \left( -{\frac {1- \left( \tanh \left( x+iy \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 \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
\right) {{\rm KummerM}\left(1,\,2,\,i\pi -2\,z\right)}{{\rm 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) }{
\left( 2\,iz+\pi \right) {\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {
1/2\,i\pi -z} \right) {{\rm e}^{2\,z-1/2\,i\pi }}}}</math>
*<math>tanhc(z)={\frac {i{{\rm \bf WhittakerM}\left(0,\,1/2,\,2\,z\right)}}{
{{\rm \bf WhittakerM}\left(0,\,1/2,\,i\pi -2\,z\right)}z}}</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]]
|}
==References==
<references/>
[[Category:special functions]]
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