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Legendre polynomials

Legendre functions are solutions to Legendre's differential equation:

<math>{d \over dx} \left[ (1-x^2) {d \over dx} P(x) \right] + n(n+1)P(x) = 0</math>

They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates.

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.

Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' Formula:

<math>P_n(x) = (2^n n!)^{-1} {d^n \over dx^n } \left[ (x^2 -1)^n \right] </math>

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:

<math>\int_{-1}^{1} P_m(x) P_n(x)\,dx = {2 \over {2n + 1}} \delta_{mn}</math>

(where δmn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise).

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:

n <math>P_n(x)</math>
0 1
<math>1</math> <math>x</math>
2 <math>(1/2)(3x^2-1)</math>
3 <math>(1/2)(5x^3-3x)</math>
4 <math>(1/8)(35x^4-30x^2+3)</math>
5 <math>(1/8)(63x^5-70x^3+15x)</math>

The graphs of these polynomials are shown below:

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