Legendre polynomial LegendrePolynomials 2005-04-22 01:25:54 2008-02-29 13:19:37 Definition Rodrigues's Formula Legendre's Differential Equation % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The \emph{Legendre polynomials} are a set of polynomials $\{P_n\}_{n=0}^{\infty}$ each of order $n$ that satisfy Legendre's ODE: $$\frac{d}{dx}[(1-x^2) P_n'(x)] + n(n+1)P_n(x) = 0.$$ Alternatively $P_n$ is an eigenfunction of the self-adjoint differential operator $\frac{d}{dx}(1-x^2) \frac{d}{dx}$ with eigenvalue $-n(n+1)$. 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 $[-1, 1]$ with weight function $W(x) = 1$. 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 $\langle P_n \| P_n \rangle = 2 / (2n + 1)$, 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 $P_n$ in terms of derivatives: $$P_n(x) = \frac{1}{2^n n!} \left( \frac{d}{dx} \right)^n (x^2 - 1)^n$$ The first few explicitly are: \begin{eqnarray*} P_0(x) &=& 1 \\ P_1(x) &=& x \\ P_2(x) &=& \frac{1}{2} (3x^2 - 1) \\ P_3(x) &=& \frac{1}{2} (5x^3 - 3x) \\ P_4(x) &=& \frac{1}{8} (35x^4 - 30x^2 + 3) \\ ... \end{eqnarray*} As all orthogonal polynomials do, these satisfy a three-term recurrence relation: $$(n+1)P_{n+1}(x) = (2n+1)xP_{n}(x) - (n)P_{n-1}(x)$$ The Legendre functions of the second kind also satisfy the Legendre ODE but are not regular at the origin. Related are the associated Legendre functions, and spherical harmonics.