imported>Sammi Brie: Adding short description: "Solutions of the Helmholtz equation" (Shortdesc helper)

2021-04-06T00:01:37Z

Adding short description: "Solutions of the Helmholtz equation" (Shortdesc helper)

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{{Short description|Solutions of the Helmholtz equation}}
'''Spheroidal wave functions''' are solutions of the [[Helmholtz equation]] that are found by writing the equation in spheroidal coordinates and applying the technique of [[separation of variables]], just like the use of [[spherical coordinates]] lead to [[spherical harmonics]]. They are called ''oblate spheroidal wave functions'' if [[oblate spheroidal coordinates]] are used and ''[[prolate spheroidal wave functions]]'' if [[prolate spheroidal coordinates]] are used.<ref name=Flammer1957>{{cite book
| author = Flammer, C.
| year = 1957
| title = Spheroidal wave functions
| publisher = Stanford University Press Stanford, Calif
| isbn =
}}</ref>
If instead of the Helmholtz equation, the [[Laplace equation]] is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the spheroidal harmonics. With oblate spheroidal coordinates, the solutions
are called ''oblate harmonics'' and with prolate spheroidal coordinates, ''prolate harmonics''. Both type of spheroidal harmonics
are expressible in terms of [[Legendre functions]].

==See also==
* [[Oblate spheroidal coordinates]], especially the section [[Oblate spheroidal coordinates#Oblate spheroidal harmonics|''Oblate spheroidal harmonics'']], for a more extensive discussion.
* [[Oblate spheroidal wave function]]

==References==
;Notes
{{reflist}}
;Bibliography
* C. Niven ''On the Conduction of Heat in Ellipsoids of Revolution.'' Philosophical transactions of the Royal Society of London, v. 171 p. 117 (1880)
* M. Abramowitz and I. Stegun, ''Handbook of Mathematical function'' (US Gov. Printing Office, Washington DC, 1964)
*{{dlmf|id=30|first=H. |last=Volkmer}}

{{mathapplied-stub}}

[[Category:Partial differential equations]]
[[Category:Special functions]]

Spheroidal wave function - Revision history

imported>Sammi Brie: Adding short description: "Solutions of the Helmholtz equation" (Shortdesc helper)