Laplace transforms

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

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

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

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

Interchange the order of integration to obtain

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

This is equivalent to the formula

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

which we will use to derive our result.