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Zbl 0606.33011
Dunster, T.M.
Uniform asymptotic expansions for prolate spheroidal functions with large parameters. (English)
[J] SIAM J. Math. Anal. 17, 1495-1524 (1986). ISSN 0036-1410; ISSN 1095-7154/e

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.

MSC 2000:: *33E10 Spheroidal wave functions, etc.
34E05 Asymptotic expansions (ODE)
30E15 Asymptotic representations in the complex domain

Keywords: spheroidal wave functions

Citations: Zbl 0591.34048

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