Uniform asymptotic approximations 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, using Olver's theory, and provides explicit error bounds and eigenvalue approximations.

Contribution

It introduces new uniform asymptotic formulas for Lamé and Mathieu functions with explicit error bounds, extending existing methods for large parameter regimes.

Findings

Derived uniform asymptotic formulas expressed via parabolic cylinder functions.

Provided explicit bounds for approximation errors.

Obtained asymptotic approximations for eigenvalues.

Abstract

By application of the theory for second-order linear differential equations with two turning points developed in \cite{Olver1975}, uniform asymptotic approximations are obtained for the Lam\'{e} and Mathieu functions with a large real parameter. The 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 their respective eigenvalues.