On the uniform asymptotic expansion of the Legendre functions

TL;DR

This paper develops a uniform asymptotic expansion for Legendre functions of large indices using the WKB method, deriving recurrence relations for the coefficients and comparing with Bessel function expansions.

Contribution

It introduces a new uniform expansion for Legendre functions with explicit recurrence formulas, enhancing asymptotic analysis techniques.

Findings

Derived recurrence formulas for expansion coefficients.

Compared Legendre and Bessel function expansions.

Provided insights into asymptotic behaviors of Legendre functions.

Abstract

An uniform expansion of the Legendre functions of large indices are considered by using the WKB approach. We obtain the recurrent formula for the coefficients of uniform expansion and compare them with the uniform expansion of the Bessel function.