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- Jan 26, 2012

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**Problem**: Let $f,g:[0,\infty)\rightarrow\mathbb{R}$ be two functions, and let $F(s)$ and $G(s)$ denote their Laplace Transforms. Show that

\[F(s)G(s)=\int_0^{\infty} e^{-st}h(t)\,dt\]

where $h(t) = \int_0^t f(t-\tau)g(\tau)\,d\tau$ (the convolution of $f$ with $g$).

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**Hint**:

\[F(s)G(s)=\int_0^{\infty}\int_0^{\infty}e^{-s(x+y)}f(x)g(y)\,dx\,dy\]

Make the change of variables $t=x+y$, $y=\tau$ and then change the order of integration.

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