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 P_{n}(x) is an nth-degree polynomial. It may be expressed using Rodrigues' Formula:
An alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1, x, x^{2}, ...}.
These are the first few Legendre polynomials:
n | |
0 | 1 |
2 | |
3 | |
4 | |
5 | |
6 |
The graphs of these polynomials (up to n=5) are shown below: