Convolution Theorem

Theorem
Let $\mathbb F \in \set {\R, \C}$.

Let $f: \R \to \F$ and $g: \R \to \F$ be functions.

Let their Laplace transforms $\laptrans {\map f t} = \map F s$ and $\laptrans {\map g t} = \map G s$ exist.

Then:
 * $\map F s \, \map G s = \displaystyle \laptrans {\int_0^t \map f u \, \map g {t - u} \rd u}$

Also presented as
Some sources give this as:


 * $\invlaptrans {\map F s \, \map G s} = \displaystyle \int_0^t \map f u \, \map g {t - u} \rd u$