= Region of absolute convergence of a Laplace transform
{title2=$\int_0^\infty e^{-\sigma t}|f(t)|\,dt<\infty$}
= Laplace convergence region
{c}
{synonym}
The <set> of <complex numbers> $s$ for which the displayed <integral> is finite is the region of absolute convergence of a <Laplace transform>. It depends only on the <real part> $\sigma=\operatorname{Re}s$. Once the condition holds for one $\sigma$, it holds for every larger <real part>. For the <exponential function> $e^{-2t}$ it is exactly $\operatorname{Re}s>-2$; in general, convergence on the boundary must be checked separately. An <analytic continuation> outside this <set> need not equal a convergent defining <integral>.
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