# Laplace transform

Let $f \colon \mathbb{R} \rightarrow \mathbb{C}$ be a function, then the Laplace transform of $f$ is the function defined by $$\mathscr{L}\{f\}(z) = \displaystyle\int_0^{\infty} e^{-zt}f(t) \mathrm{d}t,$$ provided this integral exists.