Ordinary Differential Equations/Legendre Equation

In mathematics, Legendre's differential equation is

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

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 ${\displaystyle x=\pm 1}$ so, in general, a series solution about the origin will only converge for ${\displaystyle \left\vert x\right\vert <1}$. When n is an integer, the solution ${\displaystyle P_{n}\left(x\right)}$ that is regular at ${\displaystyle x=1}$ is also regular at ${\displaystyle x=-1}$, and the series for this solution terminates (i.e. is a polynomial).

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

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