\begin{definition}[Definition:Laplace Transform]
Let $f: \R_{\ge 0} \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \set {\R, \C}$.
The '''Laplace transform''' of $f$, denoted $\laptrans f$ or $F$, is defined as:
:$\ds \laptrans {\map f t} = \map F s = \int_0^{\to +\infty} e^{-s t} \map f t \rd t$
whenever this improper integral converges.
If this improper integral does not converge, then $\laptrans {\map f t}$ does not exist.
Here $\laptrans f$ is a complex function of the variable $s$.
\end{definition}