Legendre polynomials: Difference between revisions

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The Legendre polynomials are solutions of the Legendre differential equation
The Legendre polynomials are solutions of the Legendre differential equation
:<math>
:<math>
\frac{d}{dx} \left[ (1-x^2) \frac{d}{dx} P(x) \right] + n(n+1)P(x)   
\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(x)}{dx^2} - 2 x \frac{dP(x)}{dx} + n(n+1) P(x) = 0.
= (1-x^2) \frac{d^2 P_n(x)}{dx^2} - 2 x \frac{dP_n(x)}{dx} + n(n+1) P_n(x) = 0.
</math>
This differential equation has another class of solutions: ''Legendre functions of the second kind'' ''Q''<sub>''n''</sub>(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
:<math>
\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}
</math>
This operator is the &theta;-dependent part of  the [[Laplace operator]] &nabla;&sup2; in spherical polar coordinates.
 
==Properties of Legendre polynomials ==
Legendre polynomials have parity (-1)<sup>''n''</sup> under ''x'' &rarr; -''x'',
:<math>P_n(-x) = (-1)^n P_n(x). \,</math>
 
The following condition normalizes the polynomials
:<math>P_n(1) = 1. \,</math>
==Recurrence Relations==
Legendre polynomials satisfy the recurrence relations
 
:<math>
(1-x^2) \frac{d}{dx} P_n = (n+1) xP_n - (n+1)  P_{n+1} = n P_{n-1} - nx P_n
</math>
 
:<math>
(n+1) P_{n+1} - (2n+1) x P_n + n P_{n-1} = 0 \,
</math>
</math>
This differential has another class of solutions: ''Legendre functions of the second kind'' ''Q''<sub>''n''</sub>(x), which are infinite series in 1/''x''. These functions are of lesser importance.

Revision as of 06:49, 21 August 2007

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 Legendre (1752–1833).

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

.

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

Using the Newton binomial and the equation

we get the explicit expression

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

where δn m 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