A simple calculation shows that the Laplace transform of $f(t)=e^{-t^2/4}$ is the function $F(p)=\sqrt{\pi}e^{p^2}\operatorname{erfc}(p)$.
I would like to find the inverse Laplace transform of $F(p)$. However, $F(p)$ is an entire function for which $p=\infty$ is an essential singularity. This means that the usual methods based on contour deformation, residue calculation or series expansion don't seem to apply. Can anyone lend me a hand with this problem?
2026-03-31 03:30:56.1774927856