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Laplace transforms

Here, we will assume basic familiarity with the Laplace transform of a function, defined by

\begin{displaymath}
\mathcal{L}[f(t)](s) := \int_0^\infty {\rm e}^{-st} f(t) {\rm d}t
{\rm .}
\end{displaymath}

Consider the integral of a function multiplied by the Laplace transform of another function, i.e.

\begin{displaymath}
\int_0^\infty g(s) \mathcal{L}[f(t)](s) {\rm d}s =
\int_0^...
...s) \int_0^\infty {\rm e}^{-st} f(t) {\rm d}t {\rm d}s{\rm .}
\end{displaymath}

Interchange the order of integration to obtain

\begin{displaymath}
\int_0^\infty \int_0^\infty {\rm e}^{-st} g(s) f(t) {\rm d}s {\rm d}t {\rm .}
\end{displaymath}

This is equivalent to the formula

\begin{displaymath}
\int_0^\infty g(s) h(s)  {\rm d}s = \int_0^\infty \int_0^\i...
...st} g(s) \mathcal{L}^{-1}[h(s)](t) {\rm d}s {\rm d}t {\rm ,}
\end{displaymath}

which we will use to derive our result.



Subsections

Alexander Frolkin 2002-02-14