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

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

It is a solution of the following differential equation:

Coshc 2D plot
Coshc'(z) 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.
Coshc abs complex 3D
Coshc Im complex 3D plot
Coshc Re complex 3D plot
Coshc'(z) Im complex 3D plot
Coshc'(z) Re complex 3D plot
Coshc'(z) abs complex 3D plot
Coshc'(x) abs density plot
Coshc'(x) Im density plot
Coshc'(x) Re density 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. Körpinar, Talat (2014). "New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime". International Journal of Theoretical Physics. 53 (9): 3208–3218. Bibcode:2014IJTP...53.3208K. doi:10.1007/s10773-014-2118-5. ISSN 0020-7748. S2CID 121715858.
  3. Nilgün Sönmez, A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry, International Mathematical Forum, 4, 2009, no. 38, 1877–1881
  4. 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.
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