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Uniform asymptotic expansions for associated Legendre functions of large order

Published online by Cambridge University Press:  12 July 2007

T. M. Dunster
Affiliation:
Department of Mathematical Sciences, San Diego State University, San Diego, CA 92182-7720, USA

Abstract

Uniform asymptotic expansions are obtained for the associated Legendre functions and , and the Ferrers functions and , as the order μ → ∞. The approximations are uniformly valid for 0 ≤ ν + ½ ≤ μ(1 − δ), where δ ∈ (0, 1) is fixed, x ∈ (−1, 1) in the real-variable case and Re z ≥ 0 in the complex-variable case. Explicit error bounds are available for all approximations. In the complex-variable case, expansions are obtained by an application of two existing general asymptotic theories to the associated Legendre differential equation: the first case (in which ν is fixed) applies to regions containing an isolated simple pole; and the second case (in which 0 ≤ ν + ½ ≤ μ(1 − δ)) applies to regions containing a coalescing turning point and double pole. In both cases, the expansions involve modified Bessel functions. In the real-variable case (in which 0 ≤ ν + ½ ≤ μ(1 − δ)), asymptotic expansions of Liouville–Green type are obtained, which involve elementary functions.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2003

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