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III.—Theory of the Whittaker Hill Equation

Published online by Cambridge University Press:  14 February 2012

Kathleen M. Urwin  and
F. M. Arscott
Kathleen M. Urwin
Affiliation:
Department of Mathematics, University of Surrey
F. M. Arscott
Affiliation:
Department of Mathematics, University of Surrey
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Extract

When the Helmholtz equation ∇2V + k2V = o is separated in the general paraboloidal coordinate system, the three ordinary differential equations obtained each take, after a suitable change of variable, the form of the Whittaker Hill equation. For the case k2 < o, a considerable amount is known about the periodic solutions of this equation. The theory for k2 < o does not carry over immediately to the case k2 < o, and so far only perturbation solutions have been obtained. This paper gives, in sections 1–5, explicit solutions for the case k2 < o, in the form of trigonometric series determined by three term recurrence relations. In sections 6, 7 some important relations and orthogonality properties are discussed, which are of particular significance in respect of application to boundary value problems. Section 8 discusses some degenerate cases, and in the Appendix an important property is established of continuity of the solutions with respect to the parameters of the equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 69 , Issue 1 , 1970 , pp. 28 - 44
Copyright
Copyright © Royal Society of Edinburgh 1970

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