= Causality in finite-time Laplace contour inversion
{title2=$e^{pt}G_T(p)$}
A <Laplace transform> $G_T(p)=\int_0^Te^{-ps}g(s)ds$ can grow exponentially as $\operatorname{Re}p\to-\infty$. At an evaluation time $t<T$, multiplying by $e^{pt}$ does not automatically permit closing the inverse contour to the left. Either retain the growth-bearing contour pieces, use a causal time convolution, or justify a deformation with the time-truncated transform. Resolvent-pole information alone is insufficient.
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