Coshc function
_plotted_in_the_complex_plane_from_-2-2i_to_2%2B2i.svg)
In mathematics, the coshc function appears frequently in papers about optical scattering,[1] Heisenberg spacetime[2] and hyperbolic geometry.[3][better source needed] For [math]\displaystyle{ z \neq 0 }[/math], it is defined as[4] [math]\displaystyle{ \operatorname{coshc}(z)=\frac {\cosh(z) }{z} }[/math]
It is a solution of the following differential equation: [math]\displaystyle{ w( z) z-2\frac {d}{dz} w (z) -z \frac {d^2}{dz^2} w (z) =0 }[/math]
_2D_plot.png)
Properties
The first-order derivative is given by
- [math]\displaystyle{ \frac {\sinh(z)}{z} - \frac {\cosh(z)}{z^2} }[/math]
The Taylor series expansion is[math]\displaystyle{ \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) }[/math]
The Padé approximant is[math]\displaystyle{ \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}} }[/math]
In terms of other special functions
- [math]\displaystyle{ \operatorname{coshc}(z) = \frac {( iz+1/2\,\pi) {\rm M}(1,2,i\pi -2z)}{e^{(i/2)\pi -z} z} }[/math], where [math]\displaystyle{ {\rm{M}}(a,b,z) }[/math] is Kummer's confluent hypergeometric function.
- [math]\displaystyle{ \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} }[/math], where [math]\displaystyle{ {\rm{HeunB}}(q, \alpha, \gamma, \delta, \epsilon ,z) }[/math] is the biconfluent Heun function.
- [math]\displaystyle{ \operatorname{coshc}(z)= \frac {-i(2\,iz+\pi) {{\rm \mathbf WhittakerM}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi) z} }[/math], where [math]\displaystyle{ {\rm{WhittakerM}}(a,b,z) }[/math] is a Whittaker function.
Gallery
_Im_complex_3D_plot.png)
_Re_complex_3D_plot.png)
_abs_complex_3D_plot.png)
_abs_density_plot.JPG)
_Im_density_plot.JPG)
_Re_density_plot.JPG)
See also
References
- ↑ den Outer, P. N.; Lagendijk, Ad; Nieuwenhuizen, Th. M. (1993-06-01). "Location of objects in multiple-scattering media" (in en). Journal of the Optical Society of America A 10 (6): 1209. doi:10.1364/JOSAA.10.001209. ISSN 1084-7529. Bibcode: 1993JOSAA..10.1209D. https://opg.optica.org/abstract.cfm?URI=josaa-10-6-1209.
- ↑ Körpinar, Talat (2014). "New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime" (in en). International Journal of Theoretical Physics 53 (9): 3208–3218. doi:10.1007/s10773-014-2118-5. ISSN 0020-7748. Bibcode: 2014IJTP...53.3208K. http://link.springer.com/10.1007/s10773-014-2118-5.
- ↑ 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" (in en). Journal of Scientific Computing 53 (3): 552–568. doi:10.1007/s10915-012-9588-5. ISSN 0885-7474.
 |
