Definition:Laplace Transform

Definition
Let $f \left({t}\right): S \to \C$ or $S \to \R$ be a function of a real variable $t$, where $\left [{t \,.\,.\, +\infty} \right) \subseteq S \subseteq \R, t \ge 0$.

The Laplace transform of $f$, denoted $\mathcal L \left({f}\right)$ or $F$, is defined as:


 * $\displaystyle \mathcal L \left\{{f\left({t}\right)}\right\} = F \left({s}\right) = \int_0^{\to +\infty} e^{-st}f\left({t}\right) \ \mathrm d t$

wherever this improper integral exists.

Here $\mathcal L \left({f}\right)$ is a complex function of the variable $s$.

However, it was who advanced sophisticated applications of this transform in the solutions of differential equations.