Fourier series

Common forms

In short, we could loosely say that if a function defined on [-l, l] can be written as S(x) below we call the formula Fourier series.
S(x)=\frac{a_0}{2}+\sum\limits_{n=1}^{\infty}(a_n·\cos\frac{n\pi}{l}x+b_n·\sin\frac{n\pi}{l}x)
a_n=\frac{1}{l}\int_{-l}^lf(x)·\cos\frac{n\pi}{l}xdx,n=0,1,2,\dots
b_n=\frac{1}{l}\int_{-l}^lf(x)·\sin\frac{n\pi}{l}xdx,n=1,2,\dots
And:

1. f(x)=S(x) when f(x) is continuous on x
2. f(x)=\frac{1}{2}[f(x-0)+f(x+0)] when x is a discontinuity point
3. f(x)=\frac{1}{2}[f(-l+0)+f(l-0)] when x=\pm l
   The rules above are known as Dirichlet convergence theorem, though I couldn’t find it on wikipedia actually.