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

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 . It is a solution of the following differential equation:

Sinhc 2D plot
Sinhc'(z) 2D plot
Sinhc integral 2D plot

Properties

The first-order derivative is given by

The Taylor series expansion is

The Padé approximant is

In terms of other special functions

  • , where is Kummer's confluent hypergeometric function.
  • , where is the biconfluent Heun function.
  • , where is a Whittaker function.
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

  1. 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.
  2. Nilgün Sönmez, A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry, International Mathematical Forum, 4, 2009, no. 38, 1877–1881
  3. 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.
  4. Weisstein, Eric W. "Sinhc Function". mathworld.wolfram.com. Retrieved 2022-11-17.
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