Convolution Theorem/Proof 1

Proof
The region in the plane over which $(1)$ is to be integrated is $\mathscr R_{t u}$ below:

Setting $t - u = v$, that is $t = u + v$, the shaded region above is transformed into the region $\mathscr R_{u v}$ the $u v$ plane:

Thus:

where $\dfrac {\map \partial {u, t} } {\map \partial {u, v} }$ is the Jacobian of the transformation:

Thus the of $(2)$ is:


 * $\displaystyle \int_{t \mathop = 0}^M \int_{u \mathop = 0}^t e^{-s t} \, \map f u \, \map g {t - u} \rd u \rd t = \int_{t \mathop = 0}^M \int_{u \mathop = 0}^{M - v} e^{-s t} \, \map f u \, \map g v \rd u \rd v$