Theorem 1For 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-definedfunction from
$\{t \in \mathbb{C} \mid \Re t > t_0 \}$ to $\mathbb{C}$ . Furthermore, the Laplace transform function is analytic.
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