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#REDIRECT [[Hyperbolic functions]]
In mathematics, the '''Coshc function''' appears frequently in papers about optical scattering,<ref>PN Den Outer, TM Nieuwenhuizen, A Lagendijk, Location of objects in multiple-scattering media, JOSA A, Vol. 10, Issue 6, pp. 1209–1218 (1993)</ref> Heisenberg Spacetime<ref>T Körpinar, New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime, International Journal of Theoretical Physics, 2014 Springer</ref> and hyperbolic geometry.<ref>Nilgün Sönmez, A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry, International Mathematical Forum, 4, 2009, no. 38, 1877 1881</ref> It is defined as<ref>JHM ten Thije Boonkkamp, J van Dijk, L Liu, Extension of the complete flux scheme to systems of conservation laws, J Sci Comput (2012) 53:552–568, DOI 10.1007/s10915-012-9588-5</ref><ref>Weisstein, Eric W. "Coshc Function." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CoshcFunction.html{{Dead link|date=July 2019 |bot=InternetArchiveBot |fix-attempted=yes }}</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]]
;Imaginary part in complex plane
:<math> \operatorname{Im} \left( \frac {\cosh(x+iy) }{x+iy} \right) </math>
;Real part in complex plane
:<math> \operatorname{Re} \left( \frac {\cosh(x+iy) }{x+iy} \right) </math>
;absolute magnitude
:<math> \left| \frac {\cosh(x+iy)}{x+iy} \right| </math>
;First-order derivative
:<math> \frac {\sinh(z)}{z} - \frac {\cosh(z)}{z^2} </math>
;Real part of derivative
:<math> -\operatorname{Re} \left( -\frac {1- (\cosh(x+iy))^2}{x+iy} +\frac{\cosh(x+iy)}{(x+iy)^2} \right) </math>
;Imaginary part of derivative
*<math>-\operatorname{Im} \left( -\frac {1-(\cosh(x+iy))^2}{x+iy} + \frac {\cosh(x+iy)}{(x+iy)^2} \right) </math>
;absolute value of derivative
:<math> \left| -\frac{1-(\cosh(x+iy))^2}{x+iy}+\frac {\cosh(x+iy)}{(x+iy)^2} \right| </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>
*<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>
* <math>\operatorname{Coshc}(z)= \frac {-i(2\,iz+\pi) {{\rm \mathbf WhittakerM}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi) z}</math>
==Series expansion==
<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>
==Padé approximation==
<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>
==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]]
|
|}
{|
|[[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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