$$ f(x+L)=f(x) $$
$$ f(x)=\displaystyle\sum_{n=-\infty}^{\infty}c_{n}{\rm e}^{ik_{n}x} $$
$$ k_{n}=\displaystyle\frac{2\pi}{L}n,\,\,\,(n=0,\,\pm 1,\,\pm 2,\,\pm 3,\,\pm 4,\,\cdots) $$
$$ c_{n}=\displaystyle\frac{1}{L}\int_{0}^{L}f(x){\rm e}^{-ik_{n}x}{\rm d}x $$