Parseval's Theorem

Theorem
Let $f$ be a function square-integrable over $\left[-\pi \ldots \pi\right]$ given by the Fourier series,


 * $\displaystyle f(x) = \sum_{n=-\infty}^\infty c_n e^{inx}$

Where,


 * $\displaystyle c_n = \frac 1 {2\pi} \int_{-\pi}^\pi f(t) e^{-int} \mathrm dt$

Then,


 * $\displaystyle \frac 1 {2\pi} \int_{-\pi}^\pi |f(x)|^2 \mathrm dx = \sum_{n=-\infty}^\infty |c_n|^2$