Sinhc function

In mathematics, the sinhc function appears frequently in papers about optical scattering,^[1] and hyperbolic geometry.^[2] For $z\neq 0$ , it is defined as^[3]^[4]

\operatorname {sinhc} (z)={\frac {\sinh(z)}{z}}

The cardinal hyperbolic sine function sinhc(z) plotted in the complex plane from -2-2i to 2+2i

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 $\sin x/x$ . It is a solution of the following differential equation:

w(z)z-2\,{\frac {d}{dz}}w(z)-z{\frac {d^{2}}{dz^{2}}}w(z)=0

Sinhc 2D plot

Sinhc'(z) 2D plot

Sinhc integral 2D plot

Properties

The first-order derivative is given by

\operatorname {sinhc} ^{\prime }(z)={\frac {\cosh(z)}{z}}-{\frac {\sinh(z)}{z^{2}}}

The Taylor series expansion is

\sum _{i=0}^{\infty }{\frac {z^{2i}}{(2i+1)!}}.

The Padé approximant is

\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}

In terms of other special functions

$\operatorname {sinhc} (z)={\frac {{\rm {KummerM}}(1,\,2,\,2\,z)}{e^{z}}}$ , where ${\rm {KummerM}}(a,b,z)$ is Kummer's confluent hypergeometric function.
$\operatorname {sinhc} (z)={\frac {\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {z}}\right)}{e^{z}}}$ , where ${\rm {HeunB}}(q,\alpha ,\gamma ,\delta ,\epsilon ,z)$ is the biconfluent Heun function.
$\operatorname {sinhc} (z)=1/2\,{\frac {{\rm {WhittakerM}}(0,\,1/2,\,2\,z)}{z}}$ , where ${\rm {WhittakerM}}(a,b,z)$ is a Whittaker function.

Gallery

Sinhc abs complex 3D

Sinhc Im complex 3D plot

Sinhc Re complex 3D plot

Sinhc'(z) Im complex 3D plot

Sinhc'(z) Re complex 3D plot

Sinhc'(z) abs complex 3D plot

Sinhc abs plot

Sinhc Im plot

Sinhc Re plot

Sinhc'(z) Im plot

Sinhc'(z) abs plot

Sinhc'(z) Re plot

See also

References

↑ den Outer, P. N.; Lagendijk, Ad; Nieuwenhuizen, Th. M. (1993-06-01). "Location of objects in multiple-scattering media". Journal of the Optical Society of America A. 10 (6): 1209. Bibcode:1993JOSAA..10.1209D. doi:10.1364/JOSAA.10.001209. ISSN 1084-7529.
↑ 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". Journal of Scientific Computing. 53 (3): 552–568. doi:10.1007/s10915-012-9588-5. ISSN 0885-7474. S2CID 8455136.
↑ Weisstein, Eric W. "Sinhc Function". mathworld.wolfram.com. Retrieved 2022-11-17.

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