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Uniform asymptotic expansions for prolate spheroidal functions with large parameters. (English) Zbl 0606.33011
By application of the theory for second order linear differential equations with a turning point and a regular (double pole) singularity developed by {\it W. G. C. Boyd} and the author [ibid. 17, 422-450 (1986; Zbl 0591.34048)] uniform asymptotic expansions are obtained for prolate spheroidal functions for large $\gamma$. The results are uniformly valid for $0\le \mu\sp 2/\gamma\sp 2\le 1+A$ and for $A'\le \lambda /\gamma\sp 2\le A''$, where A, A’ and A” are arbitrary real constants such that $0\le A<A'\le A''<\infty$. An asymptotic relationship between $\lambda$, $\mu$, $\gamma$ and the characteristic component $\nu$ is then derived from the approximations for the spheroidal functions. All the error terms associated with the approximations have explicit bounds given.

33E10Lamé, Mathieu, and spheroidal wave functions
34E05Asymptotic expansions (ODE)
30E15Asymptotic representations in the complex domain
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