The asymptotic expansion of legendre function of large degree and order

Abstract

New expansions for the Legendre functions and are obtained; m and n are large positive numbers, is kept fixed as is an unrestricted complex variable. Three groups of expansions are obtained. The first is in terms of exponential functions. These expansions are uniformly valid as with respect to z for all z lying in except for the strips given by. The second set of expansions is in terms of Airy functions. These expansions are uniformly valid with respect to z throughout the whole z plane cut from except for a pear-shaped domain surrounding the point z = — 1 and a strip lying immediately below the real z axis for which . The third group of expansions is in terms of Bessel functions of order m . These expansions are valid uniformly with respect to z over the whole cut z plane except for the pear-shaped domain surrounding z = — 1. No expansions have been given before for the Legendre functions of large degree and order.