Parseval's Theorem/Formulation 2

Theorem
Let $f$ be a real function which is square-integrable over the interval $\openint {-\pi} \pi$.

Let $f$ be expressed by the Fourier series:


 * $\map f x = \ds \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$

where:


 * $c_n = \ds \frac 1 {2 \pi} \int_{-\pi}^\pi \map f t e^{-i n t} \rd t$

Then:


 * $\ds \frac 1 {2 \pi} \int_{-\pi}^\pi \size {\map f x}^2 \rd x = \sum_{n \mathop = -\infty}^\infty \size {c_n}^2$