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III.—Theory of the Whittaker Hill Equation
Published online by Cambridge University Press: 14 February 2012
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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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