## The Laplace Transform

The expression “It is easy to see that…” was used many and many times by the scientist Pierre-Simon Laplace when he didn’t want to come into the details of its ideas. One of the main contributions of Laplace, the Laplace Transform, will be explained here, hopefully in an easy to see fashion.

## Definition

Let $f:\mathcal{R}\rightarrow\mathcal{R}$. Its Laplace transform is defined as:

$\displaystyle \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty}{e^{-st}f(t)dt}.$

If $f$ is piecewise continuous and there exist real positive numbers $a,K,M$ such that $|f(t)| \leq Ke^{at} \quad \forall t \geq M$, then the integral above is well defined for all values $s > a$.