Newman's Tauberian Theorem

This article is a stub. Help us out by expanding it.

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

Page

Toolbox

Search

Newman's Tauberian Theorem

Statement

Proof