Sinhc function: Difference between revisions

Revision as of 20:24, 17 November 2022

Revision as of 20:32, 17 November 2022

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[[File:Sinhc 2D plot.png|thumb|Sinhc 2D plot]]

[[File:Sinhc'(z) 2D plot.png|thumb|Sinhc'(z) 2D plot]]

[[File:Sinhc integral 2D plot.png|thumb|Sinhc integral 2D plot]]

== Properties ==
The first-order derivative is given by

:<math> \frac {\cosh(z)}{z} - \frac {\sinh(z)}{z^2} </math>

The [[Taylor series]] expansion is<math display="block">\sum_{i=0}^\infty \frac{z^{2i}}{(2i+1)!}.</math>The [[Padé approximant]] is<math display="block"> \operatorname{Sinhcsinhc} \left( z \right) = \left( 1+{\frac {53272705}{360869676}}

==In terms of other special functions==

* <math>\operatorname{Sinhc}(z)=\frac {{\rm KummerM}(1,\,2,\,2\,z)}{e^z}</math>

* <math>\operatorname{Sinhc}(z)=\frac {\operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {z} \right) }{e^z} </math>

* <math>\operatorname{Sinhc}(z)=1/2\,\frac {{{\rm WhittakerM}(0,\,1/2,\,2\,z)}}{z} </math>

==Series expansion==

<math display="block">\sum_{i=0}^\infty \frac{z^{2i}}{(2i+1)!}.</math>

==Padé approximation==

<math display="block"> \operatorname{Sinhc} \left( z \right) = \left( 1+{\frac {53272705}{360869676}}

\,{z}^{2}+{\frac {38518909}{7217393520}}\,{z}^{4}+{\frac {269197963}{

3940696861920}}\,{z}^{6}+{\frac {4585922449}{15605159573203200}}\,{z}^

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+{\frac {1029037}{346781323848960}}\,{z}^{8} \right) ^{-1}

</math>

=== In terms of other special functions ===

* <math>\operatorname{Sinhcsinhc}(z)=\frac {{\rm KummerM}(1,\,2,\,2\,z)}{e^z}</math>, where <math>{\rm{KummerM}}(a,b,z)</math> is Kummer's [[confluent hypergeometric function]].

* <math>\operatorname{Sinhcsinhc}(z)=\frac {\operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {z} \right) }{e^z} </math>, where <math>{\rm{HeunB}}(q, \alpha, \gamma, \delta, \epsilon ,z)</math> is the biconfluent [[Heun function]].

* <math>\operatorname{Sinhcsinhc}(z)=1/2\,\frac {{{\rm WhittakerM}(0,\,1/2,\,2\,z)}}{z} </math>, where <math>{\rm{WhittakerM}}(a,b,z)</math> is a [[Whittaker function]].

==Gallery==