Hostname: page-component-7479d7b7d-rvbq7 Total loading time: 0 Render date: 2024-07-11T23:25:34.840Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References to Literature

Arscott, F. M., 1964 a. Periodic Differential Equations. Oxford: Pergamon Press.Google Scholar
Arscott, F. M., 1964 b. “Paraboloidal coordinates and Laplace's equation”, Proc. Roy. Soc. Edinb., A, 66, 129–139.Google Scholar
Arscott, F. M., 1964 c. “Two parameter eigenvalue problems in differential equations”, Proc. Lond. Math. Soc., 14, 459–470.CrossRefGoogle Scholar
Arscott, F. M., 1967. “The Whittaker Hill equation and the wave equation in paraboloidal coordinates”, Proc. Roy. Soc. Edinb., A, 67, 265–276.Google Scholar
Erdélyi, A. (Ed.), 1953. Higher Transcendental Functions, vol. II. McGraw Hill.Google Scholar
Ince, E. L., 1923. “A linear differential equation with periodic coefficients”, Proc. Lond. Math. Soc., 23, 56–74.Google Scholar
Ince, E. L., 1926 a. “The real zeros of solutions of a linear differential equation with periodic coefficients”, Proc Lond. Math. Soc., 25, 53–58.CrossRefGoogle Scholar
Ince, E. L., 1926 b. Ordinary Differential Equations. London: Longmans.Google Scholar
Magnus, W., and Winkler, S., 1966. Hill's Equation New York: Wiley.Google Scholar
McLachlan, N. W., 1947. Theory and Application of Mathieu Functions. Oxford: Clarendon.Google Scholar
Meixner, J., and Schäfke, F. W., 1954. Mathieusche Funktionen und Sphäroidfunktionen. Springer Verlag.CrossRefGoogle Scholar
Urwin, K. M., 1964. “Integral equations for paraboloidal wave functions, I”, Q.Jl Math., 15, 309–315Google Scholar
Urwin, K. M., 1965. “Integral equations for paraboloidal wave functions, II”, Q.Jl Math., 16, 257–262CrossRefGoogle Scholar
Urwin, K. M., 1967. “Perturbation solutions of the Whittaker Hill equation” J. Inst. Math. Applic., 3, 169–180.CrossRefGoogle Scholar
Whittaker, E. T., 1914. “On a class of differential equations whose solutions satisfy integral equations”, Proc. Edinb. Math. Soc., 33, 14–23.CrossRefGoogle Scholar