Legendre polynomials

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In mathematics, the Legendre polynomials Pn(x) are orthogonal polynomials in the variable −1 ≤ x ≤ 1. Their orthogonality is with unit weight,

The polynomials are named after the French mathematician Adrien-Marie Legendre (1752–1833).

In physics they commonly appear as a function of a polar angle 0 ≤ θ ≤ π with x = cosθ

.

The polynomials as function of cosθ are part of the solution of the Laplace equation in spherical polar coordinates.

By the sequential Gram-Schmidt orthogonalization procedure applied to {1, x, x², x³, …} the nth degree polynomial Pn can be constructed recursively. The Gram-Schmidt procedure applies to all members of the family of orthogonal polynomials, such as Hermite polynomials, Chebyshev polynomials, etc. Further, Pn(x) has in common with the other orthogonal polynomials that it has exactly n real distinct zeroes. These zeroes are used as grid points in Gauss quadrature (numerical integration) schemes.

Rodrigues' formula

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

Using the Newton binomial and the equation

we get the explicit expression

Substitution p=n-k gives this formula a slightly different appearance

Generating function

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

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

where

Obviously the expansion makes sense only if R > r.

Normalization

The polynomials are not normalized to unity, but

where δnm is the Kronecker delta.

Differential equation

The Legendre polynomials are solutions of the Legendre differential equation

This differential equation has another class of solutions: Legendre functions of the second kind Qn(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

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,

The following condition normalizes the polynomials

Recurrence Relations

Legendre polynomials satisfy the recurrence relations

From these two relations follows easily

External link

Weisstein, Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web Resource. [1]