Table of Integrals, Series, and Products, 8th Edition

8.7–8.8 Associated Legendre Functions

8.70 Introduction

8.700 An associated Legendre function is a solution of the differential equation

1. 

$\left(1-z^2\right)\frac{d^{2}u}{d{z}^2}-2z\frac{du}{dz}+\left[\nu \left(\nu +1\right)-\frac{{\mu }^2}{1-z^2}\right]\text{?}\text{?}u=0,$

in which ν and μ are arbitrary complex constants.

This equation is a special case of (Riemann's) hypergeometric equation (see 9.151). The points

$+1,-1,\infty$

are, in general, its singular points, specifically, its ordinary branch points.

We are interested, on the one hand, in solutions of the equation that correspond to real values of the independent variable ...