Ordinary Differential Equations/Legendre Equation

< Ordinary Differential Equations

In mathematics, Legendre's differential equation is

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

They are named after Adrien-Marie Legendre. 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 equation has regular singular points at x=\pm1 so, in general, a series solution about the origin will only converge for \left\vert x \right\vert < 1. When n is an integer, the solution P_n\left(x\right) that is regular at x = 1 is also regular at x=-1, and the series for this solution terminates (i.e. is a polynomial).

The solutions for n = 0, 1, 2 \dots (with the normalization P_n\left(1\right)=1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial P_n\left(x\right) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:

P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right].