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Legendre polynomials

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In mathematics, the Legendre polynomials P_n(x) are orthogonal polynomials in the variable -1 ≤ x ≤ 1. Their orthogonality is with unit weight,

$\int_{-1}^{1} P_{n}(x) P_{n'}(x) dx = 0\quad \hbox{for}\quad n\ne n'.$

The polynomials are named after the French mathematician Legendre (1752–1833).

In physics they commonly appear as a function of a polar angle 0 ≤ θ ≤ π with x = cosθ

$\int^{\pi}_{0} P_{n}(\cos\theta) P_{n'}(\cos\theta) \sin\theta \;d\theta = 0\quad \hbox{for}\quad n\ne n'.$ .

The polynomials as function of cosθ are part of the solution of the Laplace equation in spherical polar coordinates.

By the sequential Gram-Schmidt orthogonalization procedure applied to {1, x, x², x³, …} the n^th degree polynomial P_n can be constructed recursively. The Gram-Schmidt procedure applies to all members of the family of orthogonal polynomials, such as Hermite polynomials, Chebyshev polynomials, etc. Further, P_n(x) has in common with the other orthogonal polynomials that it has exactly n real distinct zeroes. These zeroes are used as grid points in Gauss quadrature (numerical integration) schemes.

1 Rodrigues' formula
2 Generating function
3 Normalization
4 Differential equation
5 Properties of Legendre polynomials
6 Recurrence Relations
7 External link

Rodrigues' formula

The French amateur mathematician Rodrigues (1795–1851) proved the following formula

$P_n(x) = {1 \over 2^n n!} \frac{d^n(x^2 -1)^n}{dx^n} .$

Using the Newton binomial and the equation

$\frac{d^n x^m}{dx^n} = \frac{m!}{(m-n)!} x^{m-n}, \quad\hbox{for}\quad n\le m,$

we get the explicit expression

$P_n(x) = \frac{1}{2^n \, n!}\sum_{k=\lceil n/2 \rceil}^n (-1)^{n-k} {n \choose k}\frac{(2k)!}{(2k-n)!} x ^{2k-n} .$

Substitution p=n-k gives this formula a slightly different appearance

$P_n(x) = \frac{1}{2^n} \sum_{p=0}^{\lfloor n/2 \rfloor} (-1)^p \frac{(2n-2p)!}{p!(n-p)!(n-2p)!} x^{n-2p}.$

Generating function

The coefficients of hⁿ in the following expansion of the generating function are Legendre polynomials

$\frac{1}{\sqrt{1-2xh+h^2}} = \sum_{n=0} P_n(x) h^n .$

The expansion converges for |h| < 1. This expansion is useful in expanding the inverse distance between two points r and R

$\frac{1}{|\mathbf{r}-\mathbf{R}|} = \frac{1}{\sqrt{r^2 + R^2 -2rR\cos\gamma}}= \frac{1}{R} \frac{1}{\sqrt{h^2 + 1 -2hx}},$

where

$h\equiv \frac{r}{R}\quad \hbox{and}\quad x \equiv \cos\gamma \equiv \mathbf{r}\cdot\mathbf{R}/(rR).$

Obviously the expansion makes sense only if R > r.

Normalization

The polynomials are not normalized to unity, but

$\int_{-1}^{1} P_{n}(x) P_{m}(x) dx = \frac{2}{2n+1} \delta_{n m},$

where δ_nm is the Kronecker delta.

Differential equation

The Legendre polynomials are solutions of the Legendre differential equation

$\frac{d}{dx} \left[ (1-x^2) \frac{d}{dx} P_n(x) \right] + n(n+1)P_n(x) = (1-x^2) \frac{d^2 P_n(x)}{dx^2} - 2 x \frac{dP_n(x)}{dx} + n(n+1) P_n(x) = 0.$

This differential equation has another class of solutions: Legendre functions of the second kind Q_n(x), which are infinite series in 1/x. These functions are of lesser importance.

Note that the differential equation has the form of an eigenvalue equation with eigenvalue -n(n+1) of the operator

$\frac{d}{d\cos\theta} \sin^2\theta \frac{d}{d\cos\theta} = \frac{1}{\sin\theta} \frac{d}{d\theta} \sin\theta \frac{d}{d\theta}$