I'm trying to compute the inverse Laplace transform of a function of such a form
$$ g(s)=\exp\left(-\tau s\sqrt{1+\sum\frac{\gamma_i}{s+p_i}}\right) $$ where $\tau$, $\gamma_i$ and $p_i$ are all real positive numbers.
I know $-p_i$ are branch points of $g(s)$ and residues don't exist for branch points, so I cannot use the residue theorem. Then I have no idea how to proceed.
Can anyone give me any suggestions on how to deal with this problem? I will update my questions once I got any clues.
EDIT1(20190715): To evaluate the Bromwich integral, I guess I need to choose a deformed integral contour to avoid branch points and branch cuts. But now I am not sure if I'm right in saying $-p_i$ are the branch points of $g(s)$. And I don't know how to determine the branch cuts of $g(s)$.