Newman's Tauberian Theorem

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Statement

Let $f:(0,+\infty)\to\mathbb C$ be a bounded function. Assume that its Laplace transform $F(s)=\int_0^\infty f(t)e^{-st}\,dt$ (which is well-defined by this formula for $\Re s>0$) admits an analytic extension (which we'll denote by the same letter $F$) to some open domain $E$ containing the closed half-plane $\{s\in\mathbb C\,:\,\Re s\ge 0\}$. Then the integral $\int_0^\infty f(t)\,dt$ converges and its value equals $F(0)$.

Proof

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