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Uniform asymptotic expansions for oblate spheroidal functions I: positive separation parameter λ

Published online by Cambridge University Press:  14 November 2011

T. M. Dunster
T. M. Dunster
Affiliation:
Department of Mathematics, San Diego State University, San Diego, CA 92182-0314, U.S.A.
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Synopsis

Uniform asymptotic expansions are derived for solutions of the spheroidal wave equation, in the oblate case where the parameter µ is real and nonnegative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u →∞, three types of expansions are derived for oblate spheroidal functions, which involve elementary, Airy and Bessel functions. Let δ be an arbitrary small positive constant. The expansions are uniformly valid for λ/u2 fixed and lying in the interval (0,2), and for λ / u2when 0<λ/u2 < 1, and when 1 = 1≦λ/u2 < 2. The union of the domains of validity of the various expansions cover the half- plane arg (z)≦ = π/2.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 3-4 , 1992 , pp. 303 - 320
Copyright
Copyright © Royal Society of Edinburgh 1992

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