The Legendre polynomials are a set of polynomials each of order that satisfy Legendre’s ODE:
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 with weight function . As with any set of orthogonal polynomials, this can be used to generate them (up to normalization) by Gram-Schmidt orthogonalization of the monomials . The normalization used is , which makes
Rodrigues’s Formula (which can be generalized to some other polynomial sets) is a sometimes convenient form of in terms of derivatives:
The first few explicitly are:
As all orthogonal polynomials do, these satisfy a three-term recurrence relation:
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.
|Date of creation||2013-03-22 15:12:10|
|Last modified on||2013-03-22 15:12:10|
|Last modified by||mathcam (2727)|
|Defines||Legendre’s Differential Equation|