How to confirm if there exists a piecewise continuous function $f(t)$ whose Laplace transform $F(s)$ is given? I know the existence theorem for the existence of Laplace transform but don't know how to use it?
Say if $F(s)=e^{s^{2}+1}$, how can I confirm if there exist a piecewise continuous function $f(t)$ corresponding to the given $F(s)$?
Could you please explain it?
Thank you!
Not an answer
I plotted numerical integrals for $t\in[-4,4]$ $$f(t)={\mathcal {L}}^{-1}\{e^{s^2+1}\}(t)={\frac {1}{2\pi i}}\int _{ -i \infty}^{i\infty}e^{st}e^{s^2+1}\,ds$$ There should exist an inverse