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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. 1209-12181209–1218 (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--AMathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CoshcFunction.html</ref>
: <math>\operatorname{Coshc}(z)=\frac {\cosh(z) }{z}</math>
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}{dz^2} w (z) =0</math>
{2}}{d{z}^{2}}}w \left( z \right) =0</math>
[[File:Coshc 2D plot.png|thumb|Coshc 2D plot]]
*<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>
;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) {\rm M}(1,2,i\rightpi -2z)}{{\rm e}^{(i/2)\pi -z} z}
{{\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 operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {1/2\,i\pi -z} \right) } {{\rm e}^{1/2\,i\pi -z}z}
\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) {{\rm \rightmathbf WhittakerM}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi) z}
{{\rm \mathbf WhittakerM}\left(0,\,1/2,\,i\pi -2\,z\right)}}{ \left( 4\,iz+2\,\pi
\right) z}}
</math>
==Series expansion==
: <math>\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}^{911}+{\frac {1}{87178291200}z^{13}+O(z^{15}) \right)</math>
479001600}}{z}^{11}+{\frac {1}{87178291200}}{z}^{13}+O \left( {z}^{15} \right) )</math>
==Gallery==
==See also==
* [[TanhcTanc function]]
* [[SinhcTanhc function]]
▲* [[ TancSinhc function]]
==References==
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