Legendre polynomials

In mathematics, the Legendre polynomials P_n(x) are orthogonal polynomials in the variable -1 ≤ x ≤ 1. Their orthogonality 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 = cosθ

$\int _{0}^{\pi }P_{n}(\cos \theta )P_{n'}(\cos \theta )\sin \theta \;d\theta =0\quad {\hbox{for}}\quad n\neq n'.$.

By the sequential Gram-Schmidt orthogonalization procedure applied to {1, x, x², x³, …} the polynomials can be constructed.

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

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

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, but

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

where δ_nm is the Kronecker delta.

Differential equation

The Legendre polynomials are solutions of the Legendre differential equation

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

This differential equation 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.

Note that the differential equation has the form of an eigenvalue equation with eigenvalue -n(n+1) of the operator

${\frac {d}{d\cos \theta }}\sin ^{2}\theta {\frac {d}{d\cos \theta }}={\frac {1}{\sin \theta }}{\frac {d}{d\theta }}\sin \theta {\frac {d}{d\theta }}$

This operator is the θ-dependent part of the Laplace operator ∇² in spherical polar coordinates.

Properties of Legendre polynomials

Legendre polynomials have parity (-1)ⁿ under x → -x,

$P_{n}(-x)=(-1)^{n}P_{n}(x).\,$

The following condition normalizes the polynomials

$P_{n}(1)=1.\,$

Recurrence Relations

Legendre polynomials satisfy the recurrence relations

$(1-x^{2}){\frac {d}{dx}}P_{n}=(n+1)xP_{n}-(n+1)P_{n+1}=nP_{n-1}-nxP_{n}$

$(n+1)P_{n+1}-(2n+1)xP_{n}+nP_{n-1}=0\,$

From these two relations follows easily

${\frac {d{\big (}P_{n+1}-P_{n-1}{\big )}}{dx}}=(2n+1)P_{n}.$