Ordinary Differential Equations/Legendre Equation

< Ordinary Differential Equations

In mathematics, Legendre's differential equation is

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 so, in general, a series solution about the origin will only converge for . When n is an integer, the solution that is regular at is also regular at , and the series for this solution terminates (i.e. is a polynomial).

The solutions for (with the normalization ) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial is an nth-degree polynomial. It may be expressed using Rodrigues' formula: