Coshc function

In mathematics, the Coshc function appears frequently in papers about optical scattering,[1] Heisenberg Spacetime[2] and hyperbolic geometry.[3] It is defined as[4][5]

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

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

Imaginary part in complex plane

$\operatorname {Im} \left({\frac {\cosh(x+iy)}{x+iy}}\right)$

Real part in complex plane

$\operatorname {Re} \left({\frac {\cosh(x+iy)}{x+iy}}\right)$

absolute magnitude

$\left|{\frac {\cosh(x+iy)}{x+iy}}\right|$

First-order derivative

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

Real part of derivative

$-\operatorname {Re} \left(-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right)$

Imaginary part of derivative

$-\operatorname {Im} \left(-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right)$

absolute value of derivative

$\left|-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right|$

In terms of other special functions

$\operatorname {Coshc} (z)={\frac {(iz+1/2\,\pi ){\rm {M}}(1,2,i\pi -2z)}{{\rm {e}}^{(i/2)\pi -z}z}}$

$\operatorname {Coshc} (z)={\frac {1}{2}}\,{\frac {(2\,iz+\pi )\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right)}{{\rm {e}}^{1/2\,i\pi -z}z}}$

$\operatorname {Coshc} (z)={\frac {-i(2\,iz+\pi ){{\rm {\mathbf {W} hittakerM}}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi )z}}$

Series expansion

\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)

Padé approximation

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

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

References

PN Den Outer, TM Nieuwenhuizen, A Lagendijk, Location of objects in multiple-scattering media, JOSA A, Vol. 10, Issue 6, pp. 1209–1218 (1993)
T Körpinar, New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime, International Journal of Theoretical Physics, 2014 Springer
Nilgün Sönmez, A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry, International Mathematical Forum, 4, 2009, no. 38, 1877 1881
JHM ten Thije Boonkkamp, J van Dijk, L Liu, Extension of the complete flux scheme to systems of conservation laws, J Sci Comput (2012) 53:552–568, DOI 10.1007/s10915-012-9588-5
Weisstein, Eric W. "Coshc Function." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CoshcFunction.html%5B%5D

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[1] PN Den Outer, TM Nieuwenhuizen, A Lagendijk, Location of objects in multiple-scattering media, JOSA A, Vol. 10, Issue 6, pp. 1209–1218 (1993)

[2] T Körpinar, New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime, International Journal of Theoretical Physics, 2014 Springer

[3] Nilgün Sönmez, A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry, International Mathematical Forum, 4, 2009, no. 38, 1877 1881

[4] JHM ten Thije Boonkkamp, J van Dijk, L Liu, Extension of the complete flux scheme to systems of conservation laws, J Sci Comput (2012) 53:552–568, DOI 10.1007/s10915-012-9588-5

[5] Weisstein, Eric W. "Coshc Function." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CoshcFunction.html%5B%5D