Difference between revisions of "Newman's Tauberian Theorem"

Revision as of 18:05, 3 July 2006

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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 $F(0)$ .

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Difference between revisions of "Newman's Tauberian Theorem"

Revision as of 18:05, 3 July 2006

Statement

Proof