Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Uniform asymptotic expansions for associated Legendre functions of large order

T. M. Dunstera1

a1 Department of Mathematical Sciences, San Diego State University, San Diego, CA 92182-7720, USA


Uniform asymptotic expansions are obtained for the associated Legendre functions S0308210500002687inline001 and S0308210500002687inline002, and the Ferrers functions S0308210500002687inline003 and S0308210500002687inline004, as the order μ → ∞. The approximations are uniformly valid for 0 ≤ ν + ½ ≤ μ(1 − δ), where δ xs2208 (0, 1) is fixed, x xs2208 (−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.

(Received April 05 2002)

(Accepted August 01 2002)