Simplified uniform asymptotic expansions for associated Legendre and conical functions

TL;DR

This paper derives simplified, uniform asymptotic expansions for associated Legendre and conical functions valid for large parameters and complex arguments, facilitating accurate computations across various cases.

Contribution

It introduces new, explicit asymptotic expansions involving exponential and Bessel functions, covering a wide range of parameter and argument values.

Findings

Expansions are valid for complex arguments including singularities.

Coefficients are simple and explicitly obtainable.

High-accuracy computations are enabled by the new formulas.

Abstract

Asymptotic expansions are derived for associated Legendre functions of degree $\nu$ and order $\mu$, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument $z$, including the singularity $z=1$. The cases where $\nu+\frac12$ and $\mu$ are real or purely imaginary are included, which covers conical functions. The approximations involve either exponential or modified Bessel functions, along with slowly varying coefficient functions. The coefficients of the new asymptotic expansions are simple and readily obtained explicitly, allowing for computation to a high degree of accuracy. The results are constructed and rigorously established by employing certain Liouville-Green type expansions where the coefficients appear in the exponent of an exponential function.