Tanhc function: Difference between revisions

Revision as of 02:14, 8 September 2022

Revision as of 20:46, 17 November 2022

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[[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." From MathWorld--A Wolfram Web Resource. http|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]]

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[[File:Tanhc integral 3D plot.png|thumb|Tanhc integral 3D plot]]

== Properties ==

;Imaginary part in complex plane

;FirstThe first-order derivative is given by

:<math> \frac {\operatorname{Imsech} \left^2(z)}{z} - \frac {\tanh(x+iyz) }{x+iyz^2} \right) </math>

;Real part in complex plane

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}{

:<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 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>

<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>

==Padé approximation==

The [[Padé approximant]] is<math display="block"> \operatorname{Tanhctanhc} \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==