Article 6YG0W Legendre polynomials

Legendre polynomials

by
John
from John D. Cook on (#6YG0W)

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

The middle bullet point means

legendre_poly1.svg

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

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

legendre_poly.png

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 Pk(x) satisfies Legendre's differential equation; that's what motivated them.

legendre_ode_k.svg

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.

legendre_triangle.png

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

legendre_law_of_cos.svg

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.

legendre_series.svg

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

Related postsThe post Legendre polynomials first appeared on John D. Cook.
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