There is a proposition about Laplace transform, but I don't know how to prove it.
Let $f \in L^1_{loc}(\mathbb{R})$, $\operatorname{supp}(f) \subset[0, \infty)$, such that $a$ is the abscissa of absolute convergence of the Laplace transform $F$ of $f$. Then $z \mapsto F(z)$ is analytic in the half plane $\operatorname{Re}(z)>a$.
A point $a $ is called the abscissa of absolute convergence of the Laplace transform $F$ of $f$, if $a$ is the minimum real number such that $ \int_0^\infty |f(t)| e^{-Re(z)t} dt $ exists for any $z$ with $Re(z) > a$.