Legendre polynomials

The previous post mentioned Legendre polynomials. This post will give a brief introduction to these polynomials and a couple hints of how they are used in applications.

One way to define the Legendre polynomials is as follows.

• P₀(x) = 1
• P_k are orthogonal on [-1, 1].
• P_k(1) = 1 for all k >= 0.

The middle bullet point means

if m n. The requirement that each P_k is orthogonal to each of its predecessors determines P_k up to a constant, and the condition P_k(1) = 1 determines this constant.

Here's a plot of the first few Legendre polynomials.

There's an interesting pattern that appears in the white space of a graph like the one above when you plot a large number of Legendre polynomials. See this post.

The Legendre polynomial P_k(x) satisfies Legendre's differential equation; that's what motivated them.

This differential equation comes up in the context of spherical harmonics.

Next I'll describe a geometric motivation for the Legendre polynomials. Suppose you have a triangle with one side of unit length and two longer sides of lengthr andy.

You can findy in terms ofr by using the law of cosines:

But suppose you want to find 1/y in terms of a series in 1/r. (This may seem like an arbitrary problem, but it comes up in applications.) Then the Legendre polynomials give you the coefficients of the series.

Source: Keith Oldham et al. An Atlas of Functions. 2nd edition.

Related posts

• Orthogonal polynomials and Gaussian quadrature (pdf)
• Hermite polynomials and normal distributions
• Chebyshev polynomials are distorted cosines

The post Legendre polynomials first appeared on John D. Cook.