Legendre polynomials

In mathematics, the Legendre polynomials P_n(x) are orthogonal polynomials in the variable -1 ≤ x ≤ 1. Their orthonormality is with unit weight,

$\int _{-1}^{1}P_{n}(x)P_{n'}(x)dx=0\quad {\hbox{for}}\quad n\neq n'.$

The polynomials are named after the French mathematician Legendre (1752–1833). In physics they commonly appear as a function of a polar angle 0 ≤ θ ≤ π with

$x=\cos \theta \quad {\hbox{and}}\quad dx=\sin \theta d\theta .$.

By repeated Gram-Schmidt orthogonalizations the polynomials can be constructed. However this is not the most convenient way.

Rodrigues' formula

The French amateur mathematician Rodrigues (1795–1851) proved the following formula

$P_{n}(x)={1 \over 2^{n}n!}{\frac {d^{n}(x^{2}-1)^{n}}{dx^{n}}}.$

Using the Newton binomial and the equation

${\frac {d^{n}x^{m}}{dx^{n}}}={\frac {m!}{(m-n)!}}x^{m-n}\quad {\hbox{for}}\quad m\geq n$

we get the explicit expression

$P_{n}(x)={\frac {1}{2^{n}\,n!}}\sum _{k=\lceil n/2\rceil }^{n}(-1)^{n-k}{n \choose k}{\frac {(2k)!}{(2k-n)!}}x^{2k-n}.$

Generating function

The coefficients of hⁿ in the following expansion of the generating function are Legendre polynomials

${\frac {1}{\sqrt {1-2xh+h^{2}}}}=\sum _{n=0}P_{n}(x)h^{n}.$

The expansion converges for |h| < 1. This expansion is useful in expanding the inverse distance between two points r and R

${\frac {1}{|\mathbf {r} -\mathbf {R} |}}={\frac {1}{\sqrt {r^{2}+R^{2}-2rR\cos \gamma }}}={\frac {1}{R}}{\frac {1}{\sqrt {h^{2}+1-2hx}}},$

where

$h\equiv {\frac {r}{R}}\quad {\hbox{and}}\quad x\equiv \cos \gamma \equiv \mathbf {r} \cdot \mathbf {R} /(rR).$

Obviously the expansion makes sense only if R > r.

Normalization

The polynomials are not normalized to unity

$\int _{-1}^{1}P_{n}(x)P_{m}(x)dx={\frac {2}{2n+1}}\delta _{nm},$

where δ_{n m} is the Kronecker delta.

Differential equation

The Legendre polynomials are solutions of the Legendre differential equation

${\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}P(x)\right]+n(n+1)P(x)=(1-x^{2}){\frac {d^{2}P(x)}{dx^{2}}}-2x{\frac {dP(x)}{dx}}+n(n+1)P(x)=0.$

This differential has another class of solutions: Legendre functions of the second kind Q_n(x), which are infinite series in 1/x. These functions are of lesser importance.