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In mathematics, the '''Sinhcsinhc function''' appears frequently in papers about [[optical scattering]],<ref>PN{{Cite Denjournal |last=den Outer, TM|first=P. N. |last2=Lagendijk |first2=Ad |last3=Nieuwenhuizen, A|first3=Th. Lagendijk,M. |date=1993-06-01 |title=Location of objects in multiple-scattering media, JOSA|url=https://opg.optica.org/abstract.cfm?URI=josaa-10-6-1209 |journal=Journal of the Optical Society of America A, Vol.|language=en |volume=10, Issue |issue=6, pp|pages=1209 |doi=10.1364/JOSAA.10.001209 1209–1218 (1993)|issn=1084-7529}}</ref> Heisenberg Spacetimespacetime<ref>T{{Cite journal |last=Körpinar, |first=Talat |date=2014 |title=New characterizationsCharacterizations for minimizingMinimizing energyEnergy of biharmonicBiharmonic particlesParticles in Heisenberg spacetimeSpacetime |url=http://link.springer.com/10.1007/s10773-014-2118-5 |journal=International Journal of Theoretical Physics, 2014|language=en |volume=53 |issue=9 |pages=3208–3218 |doi=10.1007/s10773-014-2118-5 Springer|issn=0020-7748}}</ref> and [[hyperbolic geometry]].<ref>Nilg¨unNilgün S¨onmezSönmez, [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> ItFor <math>z \neq 0</math>, it is defined as<ref>JHM{{Cite journal |last=ten Thije Boonkkamp, |first=J. H. M. |last2=van Dijk, L|first2=J. |last3=Liu, |first3=L. |last4=Peerenboom |first4=K. S. C. |date=2012 |title=Extension of the completeComplete fluxFlux schemeScheme to systemsSystems of conservationConservation laws,Laws J|url=http://link.springer.com/10.1007/s10915-012-9588-5 Sci|journal=Journal Computof (2012)Scientific Computing |language=en |volume=53:552–568, DOI|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." From MathWorld--A Wolfram Web Resource. http|url=https://mathworld.wolfram.com/SinhcFunction |access-date=2022-11-17 |website=mathworld.htmlwolfram.com |language=en}}</ref>
<math display="block">\operatorname{Sinhcsinhc}(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>
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