Legendre polynomials
/ luh-ZHAHN-druh /
Suppose you want to approximate a function on the interval from minus 1 to 1 by simple polynomials, but you want the building blocks to be mutually independent in an averaged sense — no overlap, so each one captures genuinely new information. The Legendre polynomials are exactly that set: the simplest family of polynomials that are orthogonal on the interval minus 1 to 1, the natural coordinate axes for functions on a sphere.
They are the polynomials P_0 = 1, P_1 = x, P_2 = (3 x^2 minus 1)/2, P_3 = (5 x^3 minus 3 x)/2, and so on, with P_n of degree n. Their defining property is orthogonality with weight 1: the integral from minus 1 to 1 of P_m(x) P_n(x) dx equals 0 whenever m is not equal to n. They can be generated cheaply by Rodrigues' formula, P_n(x) = (1 over 2^n n!) times the n-th derivative of (x^2 minus 1)^n, by a three-term recurrence (n+1) P_{n+1} = (2n+1) x P_n minus n P_{n-1}, or from a generating function. Each is normalised by P_n(1) = 1.
Legendre polynomials are the angular part of every problem with spherical symmetry: the multipole expansion of a gravitational or electrostatic potential is literally a sum of Legendre polynomials in the cosine of the polar angle, and the spherical harmonics that organise the hydrogen atom and the cosmic microwave background are built from them. Here they are catalogued as a family of orthogonal functions; the differential equation they solve and their role in eigenfunction expansions belong to neighbouring topics.
Check orthogonality of P_1 = x and P_2 = (3 x^2 minus 1)/2: integral from minus 1 to 1 of x times (3 x^2 minus 1)/2 dx = 0, because the integrand is odd. Their non-overlap is what lets a Legendre series read off one coefficient at a time.
Orthogonality on the unit interval is the property that makes Legendre polynomials a usable basis for expansion.
These orthogonal polynomials are distinct from the associated Legendre functions P_n^m and from Legendre functions of the second kind Q_n, which carry the same name but are different objects.