Definition:Laplace Transform

Definition
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 $s$.