Complex Fourier series
= Complex Fourier series
{title2=$f\sim\sum_{n\in\mathbb Z}c_ne^{in\pi x/L}$}
The complex form of a <Fourier series> for period $2L$ uses coefficients $c_n=(2L)^{-1}\int_{-L}^Lf(x)e^{-in\pi x/L}dx$ 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 $c_{-n}=\overline{c_n}$.