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On certain expansions of the solutions of the general Lamé equation

Published online by Cambridge University Press:  24 October 2008

A. Erdélyi
A. Erdélyi
Affiliation:
Mathematical InstituteThe UniversityEdinburgh 1
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Extract

1. In this paper I shall deal with the solutions of the Lamé equation

when n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the form

where θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.

It is easy to obtain the system of recurrence relations

for the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for which

k′ being the principal value of (1−k2)½

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 38 , Issue 4 , November 1942 , pp. 364 - 367
Copyright
Copyright © Cambridge Philosophical Society 1942

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References

REFERENCES

(1)Erdélyi, A.On certain expansions of the solutions of Mathieu's differential equation. Proc. Cambridge Phil. Soc. 38 (1942), 28–33.CrossRefGoogle Scholar
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(3)Hobson, E. W.The theory of spherical and ellipsoidal harmonics (Cambridge, 1931).Google Scholar
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(5)Whittaker, E. T. and Watson, G. N.Modern analysis (Cambridge, 1927).Google Scholar