Parseval's Theorem/Formulation 2

Theorem
Let $f$ be a real function which is square-integrable over the interval $\left[{-\pi \,.\,.\, \pi}\right]$.

Let $f$ be expressed by the Fourier series:


 * $\displaystyle f \left({x}\right) = \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$

where:


 * $\displaystyle c_n = \frac 1 {2 \pi} \int_{-\pi}^\pi f \left({t}\right) e^{-i n t} \, \mathrm d t$

Then:


 * $\displaystyle \frac 1 {2 \pi} \int_{-\pi}^\pi \left\lvert{f \left({x}\right)}\right\rvert^2 \, \mathrm d x = \sum_{n \mathop = -\infty}^\infty \left\lvert{c_n}\right\rvert^2$