Fourier-Legendre Series

Because the Legendre polynomials form a complete orthogonal system over the interval with respect to the weighting function , any function may be expanded in terms of them as

(1)

To obtain the coefficients in the expansion, multiply both sides by and integrate

(2)

But the Legendre polynomials obey the orthogonality relationship

(3)

where is the Kronecker delta, so

			(4)
			(5)

and

(6)

For example, for , the first few terms of the Fourier-Legendre series are

(7)