The Asymptotic Expansion of Legendre Functions of Large Degree and Order

Abstract

New expansions for the Legendre functions P_n^(-m)(z) and Q_n^(-m)(z) are obtained; m and n are large positive numbers, 0 < m < n and α = m/(n +1/2) is kept fixed as n→ ∞; z 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 n→ ∞ with respect to z for all z lying in Rz ≥ 0 except for the strips given by | lz |< ∂, Rz < β + ∂, where ∂ > 0 and β = √(1 - α^2). 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 +1 to -∞ except for a pear-shaped domain surrounding the point z = -1 and a strip lying immediately below the real z axis for which |Rzl < β + ∂, 0 ≥ lz> -∂. 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.

Additional details