Uniform Asymptotic Expansions for Prolate Spheroidal Functions with Large Parameters

Abstract

By application of the theory for second order linear differential equations with a turning point and a regular (double pole) singularity developed by Boyd and Dunster (this Journal, 17 (1986), pp. 422–450) uniform asymptotic expansions are obtained for prolate spheroidal functions for large $\gamma$. The results are uniformly valid for ${{0 \leqslant \mu ^2 } / {\gamma ^2 \leqslant 1 + A}}$ and for ${{A' \leqslant \lambda } / {\gamma ^2 \leqslant A''}}$, where A, $A'$ and $A''$ are arbitrary real constants such that $0 \leqq A < A' \leqq A'' < \infty $. An asymptotic relationship between $\lambda $, $\mu $, $\gamma $ and the characteristic exponent $\nu $ is then derived from the approximations for the spheroidal functions. All the error terms associated with the approximations have explicit bounds given.