Does an inverse Laplace transform for $\hat{F}(s)=e^{-is}$ exist? If not, why?

Question

Does an inverse Laplace transform for $\hat{F}(s)=e^{-is}$ exist? If not, why? The Bromwich integral is not covered in my course so I can't use it. I'm hoping and guessing that the answer is simple! Thanks.

Answer 1

No, it does not exist. If $f(t)$ is any measurable function such that $|f(t)| e^{-st}$ is integrable for some $s$, then its Laplace transform $F(s) = \int_0^\infty f(t) e^{-st}\; dt \to 0$ as $s \to +\infty$ (you can use the Lebesgue Dominated Convergence Theorem). But $\exp(-is)$ does not have a limit as $s \to +\infty$, so it can't be a Laplace transform.