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On addition theorems for spheroidal harmonics with some applications

Published online by Cambridge University Press:  26 February 2010

R. Shail
R. Shail
Affiliation:
Department of Mathematics, University of Surrey, London, S.W.11.
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Extract

In many physical problems it is necessary to express a solution of Laplace's equation relative to one set of coordinates in terms of harmonics relative to another set. We term such a relationship an addition or shift formula. Well-known examples are the formulae for spherical harmonics [Hobson 1, p. 139] and cylindrical harmonics [Watson 2, p. 360]. In this paper we shall consider the problem for oblate spheroidal coordinate systems and obtain some addition theorems for the corresponding harmonics. These addition theorems are used to write down “two-centred” expansions of the Coulomb Green's function, and by a limiting process a new form of the “one-centred” expansion is obtained in a non-orthogonal coordinate system, closely related to oblate spheroidal coordinates. This expansion is applied to the evaluation of the Coulomb (or gravitational) energy of spheroidal distributions of charge (or mass) in which the surfaces of constant density are concentric similar spheroids, a situation which occurs in both nuclear physics and cosmology [Carlson 3]

Type
Research Article
Information
Mathematika , Volume 14 , Issue 2 , December 1967 , pp. 132 - 141
Copyright
Copyright © University College London 1967

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References

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