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#REDIRECT [[Hyperbolic functions]]
[[File:The cardinal hyperbolic cosine function coshc(z) plotted in the complex plane from -2-2i to 2+2i.svg|alt=The cardinal hyperbolic cosine function coshc(z) plotted in the complex plane from -2-2i to 2+2i|thumb|The cardinal hyperbolic cosine function coshc(z) plotted in the complex plane from -2-2i to 2+2i]]
In mathematics, the '''coshc function''' appears frequently in papers about [[optical scattering]],<ref>{{Cite journal |last1=den Outer |first1=P. N. |last2=Lagendijk |first2=Ad |last3=Nieuwenhuizen |first3=Th. M. |date=1993-06-01 |title=Location of objects in multiple-scattering media |url=https://opg.optica.org/abstract.cfm?URI=josaa-10-6-1209 |journal=Journal of the Optical Society of America A |language=en |volume=10 |issue=6 |pages=1209 |doi=10.1364/JOSAA.10.001209 |bibcode=1993JOSAA..10.1209D |issn=1084-7529}}</ref> Heisenberg spacetime<ref>{{Cite journal |last=Körpinar |first=Talat |date=2014 |title=New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime |url=http://link.springer.com/10.1007/s10773-014-2118-5 |journal=International Journal of Theoretical Physics |language=en |volume=53 |issue=9 |pages=3208–3218 |doi=10.1007/s10773-014-2118-5 |bibcode=2014IJTP...53.3208K |s2cid=121715858 |issn=0020-7748}}</ref> and [[hyperbolic geometry]].<ref>Nilgün Sönmez, [http://www.m-hikari.com/imf-password2009/37-40-2009/sonmezIMF37-40-2009.pdf A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry], International Mathematical Forum, 4, 2009, no. 38, 1877–1881</ref>{{Better source needed|reason=Predatory open-access journal|date=November 2022}} For <math>z \neq 0</math>, it is defined as<ref>{{Cite journal |last1=ten Thije Boonkkamp |first1=J. H. M. |last2=van Dijk |first2=J. |last3=Liu |first3=L. |last4=Peerenboom |first4=K. S. C. |date=2012 |title=Extension of the Complete Flux Scheme to Systems of Conservation Laws |journal=Journal of Scientific Computing |language=en |volume=53 |issue=3 |pages=552–568 |doi=10.1007/s10915-012-9588-5 |s2cid=8455136 |issn=0885-7474|doi-access=free }}</ref>
<math display="block">\operatorname{coshc}(z)=\frac {\cosh(z) }{z}</math>
{{Rcat shell|
It is a solution of the following differential equation:
{{R to related topic}}
<math display="block">w( z) z-2\frac {d}{dz} w (z) -z \frac {d^2}{dz^2} w (z) =0</math>
[[File:Coshc 2D plot.png|thumb|Coshc 2D plot]]
[[File:Coshc'(z) 2D plot.png|thumb|Coshc'(z) 2D plot]]
== Properties ==
The first-order derivative is given by
:<math> \frac {\sinh(z)}{z} - \frac {\cosh(z)}{z^2} </math>
The [[Taylor series]] expansion is<math display="block">\operatorname{coshc} z \approx \left(z^{-1}+\frac {1}{2}z+\frac {1}{24}z^3+\frac {1}{720}z^5+\frac {1}{40320}z^7+\frac {1}{3628800}z^9+\frac {1}{479001600}z^{11}+\frac {1}{87178291200}z^{13}+O(z^{15}) \right)</math>
The [[Padé approximant]] is<math display="block">\operatorname{Coshc} \left( z \right) ={\frac {23594700729600+11275015752000\,{
z}^{2}+727718024880\,{z}^{4}+13853547000\,{z}^{6}+80737373\,{z}^{8}}{
147173\,{z}^{9}-39328920\,{z}^{7}+5772800880\,{z}^{5}-522334612800\,{z
}^{3}+23594700729600\,z}}
</math>
=== In terms of other special functions ===
* <math>\operatorname{coshc}(z) = \frac {( iz+1/2\,\pi) {\rm M}(1,2,i\pi -2z)}{e^{(i/2)\pi -z} z} </math>, where <math>{\rm{M}}(a,b,z)</math> is Kummer's [[confluent hypergeometric function]].
*<math>\operatorname{coshc}(z)=\frac{1}{2}\,\frac {(2\,iz+\pi) \operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {1/2\,i\pi -z} \right) } {e^{1/2\,i\pi -z}z} </math>, where <math>{\rm{HeunB}}(q, \alpha, \gamma, \delta, \epsilon ,z)</math> is the biconfluent [[Heun function]].
* <math>\operatorname{coshc}(z)= \frac {-i(2\,iz+\pi) {{\rm WhittakerM}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi) z}</math>, where <math>{\rm{WhittakerM}}(a,b,z)</math> is a [[Whittaker function]].
==Gallery==
{|
|[[File:Coshc abs complex 3D plot.png|thumb|Coshc abs complex 3D]]
|[[File:Coshc Im complex 3D plot.png|thumb|Coshc Im complex 3D plot]]
|[[File:Coshc Re complex 3D plot.png|thumb|Coshc Re complex 3D plot]]
{|
|[[File:Coshc'(z) Im complex 3D plot.png|thumb|Coshc'(z) Im complex 3D plot]]
|[[File:Coshc'(z) Re complex 3D plot.png|thumb|Coshc'(z) Re complex 3D plot]]
|[[File:Coshc'(z) abs complex 3D plot.png|thumb|Coshc'(z) abs complex 3D plot]]
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|}
{|
|[[File:Coshc'(x) abs density plot.JPG|thumb|Coshc'(x) abs density plot]]
|[[File:Coshc'(x) Im density plot.JPG|thumb|Coshc'(x) Im density plot]]
|[[File:Coshc'(x) Re density plot.JPG|thumb|Coshc'(x) Re density plot]]
|}
==See also==
* [[Tanc function]]
* [[Tanhc function]]
* [[Sinhc function]]
==References==
<references/>
[[Category:Special functions]]
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