Legendre polynomials: Difference between revisions

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imported>Paul Wormer
(Start of Legendre polynomials (from scratch))
 
imported>Paul Wormer
(Integral over cos theta in the lead)
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\int_{-1}^{1} P_{n}(x) P_{n'}(x) dx = 0\quad \hbox{for}\quad n\ne n'.
\int_{-1}^{1} P_{n}(x) P_{n'}(x) dx = 0\quad \hbox{for}\quad n\ne n'.
</math>
</math>
The polynomials are named after the French mathematician [[Legendre]] (1752–1833). In [[physics]] they commonly appear as a function of a polar angle 0 &le; &theta; &le; &pi; with  
The polynomials are named after the French mathematician [[Legendre]] (1752–1833).
:<math> x = \cos\theta\quad \hbox{and}\quad dx = \sin\theta d\theta.  
 
In [[physics]] they commonly appear as a function of a polar angle 0 &le; &theta; &le; &pi; with ''x'' = cos&theta;
:<math>  
\int^{\pi}_{0} P_{n}(\cos\theta) P_{n'}(\cos\theta) \sin\theta \;d\theta = 0\quad \hbox{for}\quad n\ne n'.
</math>.
</math>.
By repeated  [[Gram-Schmidt orthogonalization]]s the polynomials can be constructed. However this is not the most convenient way.
By repeated  [[Gram-Schmidt orthogonalization]]s the polynomials can be constructed.
 
==Rodrigues' formula==
==Rodrigues' formula==
The French amateur mathematician Rodrigues (1795–1851) proved the following formula
The French amateur mathematician Rodrigues (1795–1851) proved the following formula

Revision as of 05:41, 21 August 2007

In mathematics, the Legendre polynomials Pn(x) are orthogonal polynomials in the variable -1 ≤ x ≤ 1. Their orthonormality 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 repeated Gram-Schmidt orthogonalizations 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 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.