Laplace Transform of Periodic Function

Theorem
Let $f: \R \to \R$ be a real function. Let $f$ be periodic, that is:


 * $\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$

Then:


 * $\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$

where $\laptrans {\map f t}$ denotes the Laplace transform.