= Newman Tauberian theorem
{c}
{title2=$\int_0^\infty f(t)\,dt=F(0)$}
For bounded locally integrable $f$ on the nonnegative real line, suppose its <Laplace transform> $F$ extends holomorphically to an open neighborhood of the closed right half-plane. Then the ordinary improper integral converges to $F(0)$. <Contour damping for bounded Laplace transforms> bounds the difference from the finite-interval transform by $2\|f\|_\infty/R$ in the limit for every radius $R$. Extension merely to the open right half-plane is insufficient, as the constant function and transform $1/z$ show.
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