existence of Laplace transform

Theorem 1.

For every measurable function $f\colon[0,\infty)\to\mathbb{C}$, if there exists a real number $t_{0}$ such that

 $\int_{0}^{\infty}e^{-st_{0}}|f(s)|\,ds$

converges, then the Laplace transform $\mathcal{L}(f)$ is a well-defined function from $\{t\in\mathbb{C}\mid\Re t>t_{0}\}$ to $\mathbb{C}$. Furthermore, the Laplace transform function is analytic.

Title existence of Laplace transform ExistenceOfLaplaceTransform 2013-03-22 16:31:12 2013-03-22 16:31:12 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Theorem msc 42-01