Coshc function: Difference between revisions

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[[File:The cardinal hyperbolic cosine function coshc(z) plotted in the complex plane from -2-2i to 2+2i.svg|alt=The cardinal hyperbolic cosine function coshc(z) plotted in the complex plane from -2-2i to 2+2i|thumb|The cardinal hyperbolic cosine function coshc(z) plotted in the complex plane from -2-2i to 2+2i]]

In mathematics, the '''Coshccoshc function''' appears frequently in papers about [[optical scattering]],<ref>PN{{Cite Denjournal |last=den Outer, TM|first=P. N. |last2=Lagendijk |first2=Ad |last3=Nieuwenhuizen, A|first3=Th. Lagendijk,M. |date=1993-06-01 |title=Location of objects in multiple-scattering media, JOSA|url=https://opg.optica.org/abstract.cfm?URI=josaa-10-6-1209 |journal=Journal of the Optical Society of America A, Vol.|language=en |volume=10, Issue |issue=6, pp.|pages=1209 1209–1218|doi=10.1364/JOSAA.10.001209 (1993)|issn=1084-7529}}</ref> Heisenberg Spacetimespacetime<ref>T{{Cite journal |last=Körpinar, |first=Talat |date=2014 |title=New characterizationsCharacterizations for minimizingMinimizing energyEnergy of biharmonicBiharmonic particlesParticles in Heisenberg spacetime,Spacetime |url=http://link.springer.com/10.1007/s10773-014-2118-5 |journal=International Journal of Theoretical Physics, 2014|language=en |volume=53 |issue=9 Springer|pages=3208–3218 |doi=10.1007/s10773-014-2118-5 |issn=0020-7748}}</ref> and [[hyperbolic geometry]].<ref>Nilgün Sönmez, [http://www.m-hikari.com/imf-password2009/37-40-2009/sonmezIMF37-40-2009.pdf A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry], International Mathematical Forum, 4, 2009, no. 38, 1877 18811877–1881</ref>{{Better Itsource needed|reason=Predatory open-access journal|date=November 2022}} For <math>z \neq 0</math>, it is defined as<ref>JHM{{Cite journal |last=ten Thije Boonkkamp, |first=J. H. M. |last2=van Dijk, L|first2=J. |last3=Liu, Extension|first3=L. of|last4=Peerenboom the|first4=K. completeS. fluxC. scheme|date=2012 to systems|title=Extension of conservationthe laws,Complete JFlux SciScheme Computto (2012)Systems 53:552–568,of DOIConservation Laws |url=http://link.springer.com/10.1007/s10915-012-9588-5</ref><ref>Weisstein, Eric|journal=Journal W.of "CoshcScientific Function."Computing From|language=en MathWorld—A|volume=53 Wolfram|issue=3 Web Resource. http://mathworld.wolfram.com/CoshcFunction.html{{Dead link|datepages=July 2019552–568 |botdoi=InternetArchiveBot10.1007/s10915-012-9588-5 |fixissn=0885-attempted=yes 7474}}</ref>

<math display="block">\operatorname{Coshccoshc}(z)=\frac {\cosh(z) }{z}</math>

It is a solution of the following differential equation:

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[[File:Coshc'(z) 2D plot.png|thumb|Coshc'(z) 2D plot]]

== Properties ==

;Imaginary part in complex plane

;FirstThe first-order derivative is given by

:<math> \operatorname{Im} \left( \frac {\cosh(x+iy) }{x+iy} \right) </math>

;Real part in complex plane

:<math> \operatorname{Re} \left( \frac {\cosh(x+iy) }{x+iy} \right) </math>

;absolute magnitude

:<math> \left| \frac {\cosh(x+iy)}{x+iy} \right| </math>

;First-order derivative

:<math> \frac {\sinh(z)}{z} - \frac {\cosh(z)}{z^2} </math>

The [[Taylor series]] expansion is<math display="block">\operatorname{Coshccoshc} 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>

;Real part of derivative

:<math> -\operatorname{Re} \left( -\frac {1- (\cosh(x+iy))^2}{x+iy} +\frac{\cosh(x+iy)}{(x+iy)^2} \right) </math>

;Imaginary part of derivative

:<math>-\operatorname{Im} \left( -\frac {1-(\cosh(x+iy))^2}{x+iy} + \frac {\cosh(x+iy)}{(x+iy)^2} \right) </math>

;absolute value of derivative

:<math> \left| -\frac{1-(\cosh(x+iy))^2}{x+iy}+\frac {\cosh(x+iy)}{(x+iy)^2} \right| </math>

The [[Padé approximant]] is<math display="block">\operatorname{Coshc} \left( z \right) ={\frac {23594700729600+11275015752000\,{

==In terms of other special functions==

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

*<math>\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>

* <math>\operatorname{Coshc}(z)= \frac {-i(2\,iz+\pi) {{\rm \mathbf WhittakerM}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi) z}</math>

==Series expansion==

<math display="block">\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>

==Padé approximation==

<math display="block">\operatorname{Coshc} \left( z \right) ={\frac {23594700729600+11275015752000\,{

z}^{2}[phone]\,{z}^{4}[phone]\,{z}^{6}[phone]\,{z}^{8}}{

147173\,{z}^{9}-39328920\,{z}^{7}[phone]\,{z}^{5}-522334612800\,{z

}^{3}[phone]\,z}}

</math>

=== In terms of other special functions ===

* <math>\operatorname{Coshccoshc}(z) = \frac {( iz+1/2\,\pi) {\rm M}(1,2,i\pi -2z)}{e^{(i/2)\pi -z} z} </math>, where <math>{\rm{M}}(a,b,z)</math> is Kummer's [[confluent hypergeometric function]].

*<math>\operatorname{Coshccoshc}(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>{\rm{HeunB}}(q, \alpha, \gamma, \delta, \epsilon ,z)</math> is the biconfluent [[Heun function]].

* <math>\operatorname{Coshccoshc}(z)= \frac {-i(2\,iz+\pi) {{\rm \mathbf WhittakerM}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi) z}</math>, where <math>{\rm{WhittakerM}}(a,b,z)</math> is a [[Whittaker function]].

==Gallery==