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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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8.7–8.8 Associated Legendre Functions

8.70 Introduction

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

1. 

(1z2)d2udz22zdudz+[ν(ν+1)μ21z2]??u=0,

si1254_e

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,

si1255_e

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 ...

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