Complex Fourier series ($f\sim {\sum }_{n\in \mathbb{Z}}{c}_n{e}^{in\pi x/L}$)

The complex form of a Fourier series for period $2L$ uses coefficients $c_n=(2L{)}^{-1}{\int }_{-L}^{L}f(x)e^{-in\pi x/L}dx$ and complex exponentials $e^{in\pi x/L}$. For square-integrable functions it converges in the $L^2$ norm, and the Parseval identity is $(2L{)}^{-1}\int |f{|}^2={\sum }_n|c_n{|}^2$. Real functions satisfy $c_{-n}=\overline{c_n}$.