Legendre Series

Because the Legendre Functions of the First Kind form a Complete Orthogonal Basis, any Function may be expanded in terms of them

$$f(x) = \sum_{n=0}^\infty a_nP_n(x).$$ (1)

Now, multiply both sides by $P_m(x)$ and integrate

$$\int^1_{-1} P_m(x)f(x)\,dx = \sum_{n=0}^\infty a_n \int^1_{-1} P_n(x)P_m(x)\,dx.$$ (2)

But

$$\int^1_{-1} P_n(x)P_m(x)\,dx = {2\over 2m+1} \delta_{mn},$$ (3)

where $\delta_{mn}$ is the Kronecker Delta, so

$$\int^1_{-1} P_m(x)f(x)\,dx = \sum_{n=0}^\infty a_n {2\over 2m+1} \delta_{mn} = {2\over 2m+1} a_m$$ (4)

and

$$a_m = {2m+1\over 2} \int^1_{-1} P_m(x)f(x)\,dx.$$ (5)

See also Fourier Series, Jackson's Theorem, Legendre Polynomial, Maclaurin Series, Picone's Theorem, Taylor Series