Legendre functions are solutions to Legendre's differential equation:
They are named after AdrienMarie 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 nonnegative integer, i.e. n = 0, 1, 2,... . In this case, the solutions form a polynomial sequence called the Legendre polynomials.
Each Legendre polynomial P_{n}(x) is an nthdegree polynomial. It may be expressed using Rodrigues' Formula:
An important property of the Legendre polynomials is that they are orthogonal with respect to the L^{2} inner product on the interval 1 ≤ x ≤ 1:
(where δ_{mn} 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 GramSchmidt process on the polynomials {1, x, x^{2}, ...}.
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^21)</math> 
3  <math>(1/2)(5x^33x)</math> 
4  <math>(1/8)(35x^430x^2+3)</math> 
5  <math>(1/8)(63x^570x^3+15x)</math> 
The graphs of these polynomials are shown below:
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