Coshc function: Difference between revisions

Revision as of 13:33, 18 March 2015

Revision as of 13:35, 18 March 2015

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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>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>

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

It is a solution of the following differential equation:

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;Imaginary part in complex plane

*<math> \operatorname{Im} \left( \frac {\sinhcosh(x+iy) }{x+iy} \right) </math>

;Real part in complex plane

*<math> \operatorname{Re} \left( \frac {\sinhcosh \left( x+iy \right) }{x+iy} \right) </math>

;absolute magnitude

*<math> \left| \frac {\sinhcosh(x+iy) }{x+iy} \right| </math>

;First-order derivative

*<math> \frac {1- \sinhcosh(z))^2}{z} - \frac {\sinhcosh(z)}{z^2} </math>

;Real part of derivative

*<math> -\operatorname{Re} \left( -\frac {1- (\sinhcosh(x+iy))^2}{x+iy} +\frac{\sinhcosh(x+iy)}{(x+iy)^2} \right)

</math>

;Imaginary part of derivative

*<math>-\operatorname{Im} \left( -\frac {1-(\sinhcosh(x+iy))^2}{x+iy} + \frac {\sinhcosh(x+iy)}{(x+iy)^2} \right)

</math>

;absolute value of derivative

*<math> \left| -\frac{1-(\sinhcosh(x+iy))^2}{x+iy}+\frac {\sinhcosh(x+iy)}{(x+iy)^2} \right| </math>

==In terms of other special functions==