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#REDIRECT [[Sinc_function#Sinhc]]
In mathematics, the '''sinhc function''' appears frequently in papers about [[optical scattering]],<ref>{{Cite journal |last=den Outer |first=P. N. |last2=Lagendijk |first2=Ad |last3=Nieuwenhuizen |first3=Th. M. |date=1993-06-01 |title=Location of objects in multiple-scattering media |url=https://opg.optica.org/abstract.cfm?URI=josaa-10-6-1209 |journal=Journal of the Optical Society of America A |language=en |volume=10 |issue=6 |pages=1209 |doi=10.1364/JOSAA.10.001209 |issn=1084-7529}}</ref> Heisenberg spacetime<ref>{{Cite journal |last=Körpinar |first=Talat |date=2014 |title=New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime |url=http://link.springer.com/10.1007/s10773-014-2118-5 |journal=International Journal of Theoretical Physics |language=en |volume=53 |issue=9 |pages=3208–3218 |doi=10.1007/s10773-014-2118-5 |issn=0020-7748}}</ref> and [[hyperbolic geometry]].<ref>Nilgün Sönmez, [http://www.m-hikari.com/imf-password2009/37-40-2009/sonmezIMF37-40-2009.pdf A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry], International Mathematical Forum, 4, 2009, no. 38, 1877–1881</ref>{{Better source needed|reason=Predatory open-access journal|date=November 2022}} For <math>z \neq 0</math>, it is defined as<ref>{{Cite journal |last=ten Thije Boonkkamp |first=J. H. M. |last2=van Dijk |first2=J. |last3=Liu |first3=L. |last4=Peerenboom |first4=K. S. C. |date=2012 |title=Extension of the Complete Flux Scheme to Systems of Conservation Laws |url=http://link.springer.com/10.1007/s10915-012-9588-5 |journal=Journal of Scientific Computing |language=en |volume=53 |issue=3 |pages=552–568 |doi=10.1007/s10915-012-9588-5 |issn=0885-7474}}</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Sinhc Function |url=https://mathworld.wolfram.com/ |access-date=2022-11-17 |website=mathworld.wolfram.com |language=en}}</ref>
<math display="block">\operatorname{sinhc}(z)=\frac {\sinh(z) }{z}</math>
[[File:The cardinal hyperbolic sine function sinhc(z) plotted in the complex plane from -2-2i to 2+2i.svg|alt=The cardinal hyperbolic sine function sinhc(z) plotted in the complex plane from -2-2i to 2+2i|thumb|The cardinal hyperbolic sine function sinhc(z) plotted in the complex plane from -2-2i to 2+2i]]
The sinhc function is the hyperbolic analogue of the [[sinc function]], defined by <math>\sin x/x</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>
{{Rcat shell|
[[File:Sinhc 2D plot.png|thumb|Sinhc 2D plot]]
{{R to related topic}}
[[File:Sinhc'(z) 2D plot.png|thumb|Sinhc'(z) 2D plot]]
[[File:Sinhc integral 2D plot.png|thumb|Sinhc integral 2D plot]]
;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>
;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>
;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)}{e^z}</math>
* <math>\operatorname{Sinhc}(z)=\frac {\operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {z} \right) }{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}^
{8} \right) \left( 1-{\frac {2290747}{120289892}}\,{z}^{2}+{\frac {
1281433}{7217393520}}\,{z}^{4}-{\frac {560401}{562956694560}}\,{z}^{6}
+{\frac {1029037}{346781323848960}}\,{z}^{8} \right) ^{-1}
</math>
==Gallery==
{|
|[[File:Sinhc abs complex 3D plot.png|thumb|Sinhc abs complex 3D]]
|[[File:Sinhc Im complex 3D plot.png|thumb|Sinhc Im complex 3D plot]]
|[[File:Sinhc Re complex 3D plot.png|thumb|Sinhc Re complex 3D plot]]
{|
|[[File:Sinhc'(z) Im complex 3D plot.png|thumb|Sinhc'(z) Im complex 3D plot]]
|[[File:Sinhc'(z) Re complex 3D plot.png|thumb|Sinhc'(z) Re complex 3D plot]]
|[[File:Sinhc'(z) abs complex 3D plot.png|thumb|Sinhc'(z) abs complex 3D plot]]
|
|}
{|
|[[File:Sinhc abs plot.JPG|thumb|Sinhc abs plot]]
|[[File:Sinhc Im plot.JPG|thumb|Sinhc Im plot]]
|[[File:Sinhc Re plot.JPG|thumb|Sinhc Re plot]]
|}
{|
|[[File:Sinhc'(z) Im plot.JPG|thumb|Sinhc'(z) Im plot]]
|[[File:Sinhc'(z) abs plot.JPG|thumb|Sinhc'(z) abs plot]]
|[[File:Sinhc'(z) Re plot.JPG|thumb|Sinhc'(z) Re plot]]
|}
==See also==
*[[Tanc function]]
*[[Tanhc function]]
*[[Sinhc integral]]
*[[Coshc function]]
==References==
<references/>
[[Category:Special functions]]
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