Let $f : [0, \infty) \to \mathbb{R}$ be a D4364: Real function such that

(i)	\begin{equation} Z : = \left\{ z \in \mathbb{C} : \int^{\infty}_0 f(t) e^{- t z} \, d t \neq \emptyset \right\} \end{equation}

The Laplace transform of $f$ is the D4881: Complex function \begin{equation} Z \to \mathbb{C}, \quad z \mapsto \int^{\infty}_0 f(t) e^{- t z} \, d t \end{equation}