Coshc function: Difference between revisions

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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. 1209–1218 (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>

: <math>\operatorname{Coshc}(z)=\frac {\cosh(z) }{z}</math>

It is a solution of the following differential equation:

: <math display="block">w( z) z-2\frac {d}{dz} w (z) -z \frac {d^2}{dz^2} w (z) =0</math>

: <math>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]]

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;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>

</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>

</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)}{{\rm 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) } {{\rm e}^{1/2\,i\pi -z}z} </math>

</math>

* <math>\operatorname{Coshc}(z)= \frac {-i(2\,iz+\pi) {{\rm \mathbf WhittakerM}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi) 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) } {{\rm 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>

: <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^{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}[phone]\,{z}^{4}[phone]\,{z}^{6}[phone]\,{z}^{8}}{

147173\,{z}^{9}-39328920\,{z}^{7}[phone]\,{z}^{5}-522334612800\,{z