PlanetMath: Legendre polynomial

(more info)

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Legendre polynomial

(Definition)

The Legendre polynomials are a set of polynomials $\{P_n\}_{n=0}^{\infty}$ each of order that satisfy Legendre's ODE:

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

Alternatively is an eigenfunction of the self-adjoint differential operator $\frac{d}{dx}(1-x^2) \frac{d}{dx}$ with eigenvalue .

The Legendre polynomials are also known as Legendre functions of the first kind.

By Sturm-Liouville theory, this means they're orthogonal over some interval with some weight function. In fact it can be shown that they're orthogonal on with weight function . As with any set of orthogonal polynomials, this can be used to generate them (up to normalization) by Gram-Schmidt orthogonalization of the monomials $\{x^i\}$ . The normalization used is $\langle P_n \Vert P_n \rangle = 2 / (2n + 1)$ , which makes $P_n(\pm 1) = (\pm 1)^n$

Rodrigues's Formula (which can be generalized to some other polynomial sets) is a sometimes convenient form of in terms of derivatives:

$\displaystyle P_n(x) = \frac{1}{2^n n!} \left( \frac{d}{dx} \right)^n (x^2 - 1)^n$

The first few explicitly are:

$\displaystyle P_0(x)$	$\displaystyle =$	$\displaystyle 1$
$\displaystyle P_1(x)$	$\displaystyle =$	$\displaystyle x$
$\displaystyle P_2(x)$	$\displaystyle =$	$\displaystyle \frac{1}{2} (3x^2 - 1)$
$\displaystyle P_3(x)$	$\displaystyle =$	$\displaystyle \frac{1}{2} (5x^3 - 3x)$
$\displaystyle P_4(x)$	$\displaystyle =$	$\displaystyle \frac{1}{8} (35x^4 - 30x^2 + 3)$
$\displaystyle ...$