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#REDIRECT [[Hyperbolic functions]]
{{inuse|24H}}
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>Nilg¨un S¨onmez,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</ref>
{{Rcat shell|
: <math>\operatorname{Coshc}(z)=\frac {\cosh(z) }{z}</math>
{{R to related topic}}
It is a solution of the following differential equation:
<math>w \left( z \right) z-2\,{\frac {d}{dz}}w \left( z \right) -z{\frac {d^
{2}}{d{z}^{2}}}w \left( z \right) =0</math>
[[File:Coshc 2D plot.png|thumb|Coshc 2D plot]]
[[File:Coshc'(z) 2D plot.png|thumb|Coshc'(z) 2D plot]]
[[File:Coshc integral 2D plot.png|thumb|Coshc integral 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 \left( x+iy \right) }{x+iy} \right) </math>
;absolute magnitude
*<math> \left| \frac {\cosh(x+iy) }{x+iy} \right| </math>
;First-order derivative
*<math> \frac {1- \cosh(z))^2}{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 { \left( iz+1/2\,\pi \right)
{{\rm M}\left(1,\,2,\,i\pi -2\,z\right)}}{{{\rm e}^{1/2\,i\pi -z}}z}}
</math>
*<math>\operatorname{Coshc}(z)=\frac{1}{2}\,{\frac { \left( 2\,iz+\pi \right) {\it HeunB} \left( 2,0,0,0,
\sqrt {2}\sqrt {1/2\,i\pi -z} \right) }{{{\rm e}^{1/2\,i\pi -z}}z}}
</math>
* <math>\operatorname{Coshc}(z)= {\frac {-i \left( 2\,iz+\pi \right)
{{\rm \bf M}\left(0,\,1/2,\,i\pi -2\,z\right)}}{ \left( 4\,iz+2\,\pi
\right) z}}
</math>
==Series expansion==
: <math>\operatorname{Coshc} z \approx (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 \left( {z}^{16} \right) )</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 abs plot.JPG|thumb|Coshc abs plot]]
|[[File:Coshc Im plot.JPG|thumb|Coshc Im plot]]
|[[File:Coshc Re plot.JPG|thumb|Coshc Re plot]]
|}
{|
|[[File:Coshc'(z) Im plot.JPG|thumb|Coshc'(z) Im plot]]
|[[File:Coshc'(z) abs plot.JPG|thumb|Coshc'(z) abs plot]]
|[[File:Coshc'(z) Re plot.JPG|thumb|Coshc'(z) Re plot]]
|}
==See also==
[[Tanc function]]
[[Tanhc function]]
[[Sinhc function]]
==References==
<references/>
[[Category:Special functions]]
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