Sinhc function
In mathematics, the sinhc function appears frequently in papers about optical scattering,[1] Heisenberg spacetime[2] and hyperbolic geometry.[3][better source needed] For [math]\displaystyle{ z \neq 0 }[/math], it is defined as[4][5] [math]\displaystyle{ \operatorname{sinhc}(z)=\frac {\sinh(z) }{z} }[/math]
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The sinhc function is the hyperbolic analogue of the sinc function, defined by [math]\displaystyle{ \sin x/x }[/math]. It is a solution of the following differential equation: [math]\displaystyle{ w(z) z-2\,\frac {d}{dz} w (z) -z \frac {d^2}{dz^2} w (z) =0 }[/math]
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Properties
The first-order derivative is given by
- [math]\displaystyle{ \frac {\cosh(z)}{z} - \frac {\sinh(z)}{z^2} }[/math]
The Taylor series expansion is[math]\displaystyle{ \sum_{i=0}^\infty \frac{z^{2i}}{(2i+1)!}. }[/math]The Padé approximant is[math]\displaystyle{ \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]
In terms of other special functions
- [math]\displaystyle{ \operatorname{sinhc}(z)=\frac {{\rm KummerM}(1,\,2,\,2\,z)}{e^z} }[/math], where [math]\displaystyle{ {\rm{KummerM}}(a,b,z) }[/math] is Kummer's confluent hypergeometric function.
- [math]\displaystyle{ \operatorname{sinhc}(z)=\frac {\operatorname{HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {z} \right) }{e^z} }[/math], where [math]\displaystyle{ {\rm{HeunB}}(q, \alpha, \gamma, \delta, \epsilon ,z) }[/math] is the biconfluent Heun function.
- [math]\displaystyle{ \operatorname{sinhc}(z)=1/2\,\frac {{{\rm WhittakerM}(0,\,1/2,\,2\,z)}}{z} }[/math], where [math]\displaystyle{ {\rm{WhittakerM}}(a,b,z) }[/math] is a Whittaker function.
Gallery
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See also
- Tanc function
- Tanhc function
- Sinhc integral
- Coshc function
References
- ↑ den Outer, P. N.; Lagendijk, Ad; Nieuwenhuizen, Th. M. (1993-06-01). "Location of objects in multiple-scattering media" (in en). Journal of the Optical Society of America A 10 (6): 1209. doi:10.1364/JOSAA.10.001209. ISSN 1084-7529. Bibcode: 1993JOSAA..10.1209D. https://opg.optica.org/abstract.cfm?URI=josaa-10-6-1209.
- ↑ Körpinar, Talat (2014). "New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime" (in en). International Journal of Theoretical Physics 53 (9): 3208–3218. doi:10.1007/s10773-014-2118-5. ISSN 0020-7748. Bibcode: 2014IJTP...53.3208K. http://link.springer.com/10.1007/s10773-014-2118-5.
- ↑ Nilgün Sönmez, A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry, International Mathematical Forum, 4, 2009, no. 38, 1877–1881
- ↑ ten Thije Boonkkamp, J. H. M.; van Dijk, J.; Liu, L.; Peerenboom, K. S. C. (2012). "Extension of the Complete Flux Scheme to Systems of Conservation Laws" (in en). Journal of Scientific Computing 53 (3): 552–568. doi:10.1007/s10915-012-9588-5. ISSN 0885-7474.
- ↑ Weisstein, Eric W.. "Sinhc Function" (in en). https://mathworld.wolfram.com/SinhcFunction.html.
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