Legendre function

For the most common case of integer degree, see Legendre polynomials and associated Legendre polynomials.

In mathematics, the Legendre functions P_λ, Q_λ and associated Legendre functions P^μ
_λ, Q^μ
_λ are generalizations of Legendre polynomials to non-integer degree.

File:Associated Legendre Poly.svg

Associated Legendre polynomial curves for l=5.

Differential equation

Associated Legendre functions are solutions of the general Legendre equation

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

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:

$P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{1+z}{1-z}\right]^{\mu/2} \,_2F_1 (-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2}),\qquad \text{for } \ |1-z|<2$

where $\Gamma$ is the gamma function and $_2F_1$ is the hypergeometric function.

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

$Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{e^{i\mu\pi}(z^2-1)^{\mu/2}}{z^{\lambda+\mu+1}} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right),\qquad \text{for}\ \ |z|>1.$

Integral representations

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

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

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

$P_s(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^2-1}\cos\theta\right)^s d\theta = \frac{1}{\pi}\int_0^1\left(x+\sqrt{x^2-1}(2t-1)\right)^s\frac{dt}{\sqrt{t(1-t)}},\qquad s\in\mathbb{C}$

Legendre function as characters

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

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

where

$\hat{f}(s)=\int_1^\infty f(x)P_s(x)dx,\qquad -1\leq\Re(s)\leq 0$

References

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External links

• Legendre function P on the Wolfram functions site.
• Legendre function Q on the Wolfram functions site.
• Associated Legendre function P on the Wolfram functions site.
• Associated Legendre function Q on the Wolfram functions site.