On certain expansions of the solutions of the general Lamé equation
Published online by Cambridge University Press: 24 October 2008
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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