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Here's two answered and upvoted questions that do not mention anything about a region of convergence in their premises.

Usage of inverse Laplace transform

Inverse Laplace Transform properities

It is possible for questions about the inverse Laplace transform to be answered without the asker being an expert on regions of convergence.

I have a laplace transform $$F(s) = \frac{1}{s(e^{-s}-c)}$$ and the inverse Laplace transform is

$$\frac{1}{2 \pi i} \lim_{T \rightarrow \infty} \int_{\gamma-iT}^{\gamma+iT} \frac{e^{st}}{s(e^{-s}-c)}ds$$

with $\gamma, c, t \in \mathbb{R}.$

I don't recall being formally taught how to manually evaluate an inverse laplace transform, I was taught "look stuff up in your handy laplace transform table", I'd like to know how to use the Cauchy residue theorem to evaluate this inverse laplace transform to obtain a closed-form expression in terms of $t$.

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    $F(s)$ has simple poles at $s=0$ and $s=\log\left(\frac{1}{c}\right)+2 i \pi n$ where $n\in \mathbb{Z}$, so you need to select $\gamma$ such that $\gamma>0$ and $\gamma>\log\left(\frac{1}{c}\right)$. I'm not sure there's a closed form, but Mathematica gives a series representation of the result which is presumably derived from a series representation of $F(s)$. – Steven Clark Sep 29 '22 at 23:02

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