The Legendre polynomials reference article from the English Wikipedia on 24-Jul-2004
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Legendre polynomials

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In mathematics, Legendre functions are solutions to Legendre's differential equation:


The Legendre differential equation may be solved using the standard power series method. The solution is finite (i.e. the series converges) provided |x| < 1. Furthermore, it is finite at x = ± 1 provided n is a non-negative integer, i.e. n = 0, 1, 2,... . In this case, the solutions form a polynomial sequence called the Legendre polynomials.


An important property of the Legendre polynomials is that they are orthogonal with respect to the L2 inner product on the interval -1 ≤ x ≤ 1:


An alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1, x, x2, ...}.

These are the first few Legendre polynomials:

0 1




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