Rigorous Asymptotics for the Lam\'e and Mathieu Functions and their Respective Eigenvalues with a Large Parameter

TL;DR

This paper develops uniform asymptotic approximations for Lamé and Mathieu functions with large parameters, providing explicit error bounds and eigenvalue asymptotics, confirming many existing theoretical results.

Contribution

It introduces a simplified method for deriving uniform asymptotic expansions with polynomial coefficients and recurrence relations, applicable to Lamé and Mathieu functions.

Findings

Explicit error bounds for approximations

Asymptotic expansions for eigenvalues

Confirmation of existing formal results

Abstract

By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in the first part of this paper for the Lam\'e and Mathieu functions with a large real parameter. These approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid in their respective real open intervals. In all cases explicit bounds are supplied for the error terms associated with the approximations. Approximations are also obtained for the large order behaviour for the respective eigenvalues. We restrict ourselves to a two term uniform approximation. Theoretically more terms in these approximations could be computed, but the coefficients would be very complicated. In the second part of this paper we use a simplified method to obtain uniform asymptotic expansions for these functions. The coefficients are just polynomials and satisfy simple recurrence relations. The price to pay is that these asymptotic expansions hold only in a shrinking interval as their respective parameters become large; this interval however encapsulates all the interesting oscillatory behaviour of the functions. This simplified method also gives many terms in asymptotic expansions for these eigenvalues, derived simultaneously with the coefficients in the function expansions. We provide rigorous realistic error bounds for the function expansions when truncated and order estimates for the error when the eigenvalue expansions are truncated. With this paper we confirm that many of the formal results in the literature are correct.