Definition:Laplace Transform

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

The Laplace transform of $f$, denoted $\mathcal Lf$ or $F$, is defined as:


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

wherever this improper integral exists.

Here $\mathcal L f$ is a complex function of the variable $s$.

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