Laplace Transform of Periodic Function

Theorem
Let $f: \R \to \R$ be a real function.

Let $f$ be periodic, i.e.:


 * $\exists T \in \R_{\ne 0}: \forall x \in \R: f \left({x}\right) = f \left({x + T}\right)$

Then:


 * $\displaystyle \mathcal L \left\{{f \left({t}\right)}\right\} = \frac 1 {1 - e^{-s T}} \int_0^T e^{-s t} f \left({t}\right) \ \mathrm d t$

where $\mathcal L \left\{{f \left({t}\right)}\right\}$ denotes the Laplace transform.