The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

The Laplace transform of a function f(t) is defined as:

$$ \mathcal{L[f(t)]}=\int_0^{\infty}f(t)e^{-st}dt $$

Denoted $ \mathcal{L[f(t)]} $, it is a linear operator of a function $f(t)$ with a real argument t that transforms it to a function $\hat{f}(s)$ with a complex argument $s$. This transformation is essentially bijective for the majority of practical uses; the respective pairs of $f(t)$ and $\hat{f}(s)$ are matched in tables.

The Laplace transform has the useful property that many relationships and operations over the originals $f(t)$ correspond to simpler relationships and operations over the images $\hat{f}(s)$.

It is named for Pierre-Simon Laplace, who introduced the transform in his work on probability theory.