Fourier Series

The fourier series of a function f(x) on the interval [-L,L] is

\text{Fourier series} = a_0 + \sum^\infty_{n=1}a_n\cos\frac{n\pi x}{L} + \sum^\infty_{n=1}b_n\sin\frac{n\pi x}{L}

Where the coeffiencts are given by

a_0 = \frac{1}{2L}\int^L_{-L}f(x)dx\\ a_n = \frac{1}{L}\int^L_{-L}f(x)\cos\frac{n\pi x}{L}dx\\ b_n = \frac{1}{L}\int^{L}_{-L}f(x)\sin\frac{n\pi x}{L}dx

For odd functions, a_n = a_0 = 0. This gives us a fourier sine series.