Sinhc function: Difference between revisions

Revision as of 19:27, 27 May 2021

Revision as of 18:48, 12 December 2021

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In mathematics, the '''Sinhc 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. "Sinhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SinhcFunction.html</ref>

: <math display="block">\operatorname{Sinhc}(z)=\frac {\sinh(z) }{z}</math>

: <math>\operatorname{Sinhc}(z)=\frac {\sinh(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:Sinhc 2D plot.png|thumb|Sinhc 2D plot]]

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;Imaginary part in complex plane

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

;Real part in complex plane

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

;absolute magnitude

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

;First-order derivative

*:<math> \frac {\cosh(z)}{z} - \frac {\sinh(z)}{z^2} </math>

;Real part of derivative

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

</math>

;Imaginary part of derivative

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

</math>

;absolute value of derivative

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

==In terms of other special functions==

* <math>\operatorname{Sinhc}(z)=\frac {{\rm KummerM}(1,\,2,\,2\,z)}{{\rm e}^z}</math>

* <math>\operatorname{Sinhc}(z)=\frac {\operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {z} \right) }{{\rm e}^z} </math>

* <math>\operatorname{Sinhc}(z)=1/2\,\frac {{{\rm WhittakerM}(0,\,1/2,\,2\,z)}}{z} </math>

==Series expansion==

: <math display="block">\sum_{i=0}^\infty \frac{z^{2i}}{(2i+1)!}</math>.

==Padé approximation==

: <math display="block"> \operatorname{Sinhc} \left( z \right) = \left( 1+{\frac {53272705}{360869676}}

\,{z}^{2}+{\frac {38518909}{7217393520}}\,{z}^{4}+{\frac {269197963}{

3940696861920}}\,{z}^{6}+{\frac {4585922449}{15605159573203200}}\,{z}^