Definition:Laplace Transform

Definition
Let $f\left({t}\right)$ be a real function, where $t \ge 0$.

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 $s$ can be complex or real, depending on context.

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