Coshc function

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

\operatorname {coshc} (z)={\frac {\cosh(z)}{z}}

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

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

Coshc 2D plot

Coshc'(z) 2D plot

Properties

The first-order derivative is given by

{\frac {\sinh(z)}{z}}-{\frac {\cosh(z)}{z^{2}}}

The Taylor series expansion is

\operatorname {coshc} z\approx \left(z^{-1}+{\frac {1}{2}}z+{\frac {1}{24}}z^{3}+{\frac {1}{720}}z^{5}+{\frac {1}{40320}}z^{7}+{\frac {1}{3628800}}z^{9}+{\frac {1}{479001600}}z^{11}+{\frac {1}{87178291200}}z^{13}+O(z^{15})\right)

The Padé approximant is

\operatorname {Coshc} \left(z\right)={\frac {23594700729600+11275015752000\,{z}^{2}+727718024880\,{z}^{4}+13853547000\,{z}^{6}+80737373\,{z}^{8}}{147173\,{z}^{9}-39328920\,{z}^{7}+5772800880\,{z}^{5}-522334612800\,{z}^{3}+23594700729600\,z}}

In terms of other special functions

$\operatorname {coshc} (z)={\frac {(iz+1/2\,\pi ){\rm {M}}(1,2,i\pi -2z)}{e^{(i/2)\pi -z}z}}$ , where ${\rm {M}}(a,b,z)$ is Kummer's confluent hypergeometric function.
$\operatorname {coshc} (z)={\frac {1}{2}}\,{\frac {(2\,iz+\pi )\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right)}{e^{1/2\,i\pi -z}z}}$ , where ${\rm {HeunB}}(q,\alpha ,\gamma ,\delta ,\epsilon ,z)$ is the biconfluent Heun function.
$\operatorname {coshc} (z)={\frac {-i(2\,iz+\pi ){{\rm {\mathbf {W} hittakerM}}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi )z}}$ , where ${\rm {WhittakerM}}(a,b,z)$ is a Whittaker function.

Gallery

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

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.
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.
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.

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