# Legendre function

In mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions Pμ
λ
, Qμ
λ
are generalizations of Legendre polynomials to non-integer degree.

Associated Legendre polynomial curves for l=5.

## Differential equation

Associated Legendre functions are solutions of the general Legendre equation

${\displaystyle (1-x^{2})\,y''-2xy'+\left[\lambda (\lambda +1)-{\frac {\mu ^{2}}{1-x^{2}}}\right]\,y=0,\,}$

where the complex numbers λ and μ are called the degree and order of the associated Legendre functions, respectively. The Legendre polynomials are the associated Legendre functions of order μ=0.

This is a second order linear equation with three regular singular points (at 1, −1, and ∞). Like all such equations, it can be converted into a hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions.

## Definition

These functions may actually be defined for general complex parameters and argument:

${\displaystyle P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {1+z}{1-z}}\right]^{\mu /2}\,_{2}F_{1}(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}}),\qquad {\text{for }}\ |1-z|<2}$

where ${\displaystyle \Gamma }$ is the gamma function and ${\displaystyle _{2}F_{1}}$ is the hypergeometric function.

The second order differential equation has a second solution, ${\displaystyle Q_{\lambda }^{\mu }(z)}$, defined as:

${\displaystyle Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1.}$

A useful relation between Legendre P and Q functions is Whipple's formula.

## Integral representations

The Legendre functions can be written as contour integrals. For example,

${\displaystyle P_{\lambda }(z)=P_{\lambda }^{0}(z)={\frac {1}{2\pi i}}\int _{1,z}{\frac {(t^{2}-1)^{\lambda }}{2^{\lambda }(t-z)^{\lambda +1}}}dt}$

where the contour winds around the points 1 and z in the positive direction and does not wind around −1. For real x, we have

${\displaystyle P_{s}(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(x+{\sqrt {x^{2}-1}}\cos \theta \right)^{s}d\theta ={\frac {1}{\pi }}\int _{0}^{1}\left(x+{\sqrt {x^{2}-1}}(2t-1)\right)^{s}{\frac {dt}{\sqrt {t(1-t)}}},\qquad s\in \mathbb {C} }$

## Legendre function as characters

The real integral representation of ${\displaystyle P_{s}}$ are very useful in the study of harmonic analysis on ${\displaystyle L^{1}(G//K)}$ where ${\displaystyle G//K}$ is the double coset space of ${\displaystyle SL(2,\mathbb {R} )}$ (see Zonal spherical function). Actually the Fourier transform on ${\displaystyle L^{1}(G//K)}$ is given by

${\displaystyle L^{1}(G//K)\ni f\mapsto {\hat {f}}}$

where

${\displaystyle {\hat {f}}(s)=\int _{1}^{\infty }f(x)P_{s}(x)dx,\qquad -1\leq \Re (s)\leq 0}$