LegendrePolynomials 2006-05-22 23:05:06 2006-05-24 04:46:53 Definition % 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 Legendre polynomials generate the power series that solves Legendre's differential equation: $$ \left ( 1 - x^2 \right ) P''(x) - 2x P'(x) + n(n+1)P(x) = 0.$$ This ordinary differential equation with variable coefficients is named in honor of Adrien-Marie Legendre (1752-1833). While quite literally following in the footsteps of Laplace, he developed the Legendre polynomials in a paper on celestial mechanics. In a strange tangled web of fate, the Legendre polynomials are heavily used in electrostatics to solve Laplace's equation in spherical coordinates $$ \nabla^2 \Phi_{sph} = 0 $$ Not yet done.... below is derivative of wiki article . This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. The Legendre differential equation may be solved using the standard power series method. The solution is finite (i.e. the series converges) provided $ |x| < 1 $. Furthermore, it is finite at $x = ± 1$ provided n is a non-negative integer, i.e. $n = 0, 1, 2,...$ . In this case, the solutions form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial $P_n(x)$ is an nth-degree polynomial. It may be expressed using Rodrigues' formula: $$ P_n(x) = (2^n n!)^{-1} {d^n \over dx^n } \left[ (x^2 -1)^n \right]. $$ \subsection{References} http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Legendre.html http://astrowww.phys.uvic.ca/~tatum/celmechs.html