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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 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 γ. The results are uniformly valid for 0μ 2 /γ 2 1+A and for A ' λ/γ 2 A '' , where A, A’ and A” are arbitrary real constants such that 0A<A ' A '' <. An asymptotic relationship between λ, μ, γ and the characteristic component ν is then derived from the approximations for the spheroidal functions. All the error terms associated with the approximations have explicit bounds given.
MSC:
33E10Lamé, Mathieu, and spheroidal wave functions
34E05Asymptotic expansions (ODE)
30E15Asymptotic representations in the complex domain