### 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 ...

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