I have to invert the following Laplace Transform:
$$ F(s)=\frac{1}{s}e^{-AsG(s)} $$
where
$$ G(s)=1+B\sum_{k=1}^{N} \frac{C_k}{s+p_k} $$
The terms $A$, $B$, $C_k,k=1\cdots N$, and $p_k,k=1\cdots N$ are all positive real constants, and $N$ is a positive integer.
As evident from the graphic below, $G(s)$ (and hence the Laplace transform itself) has branch points at $s=-p_k,k=1\cdots N$, all of which lie on the negative real axis.
There is also a simple pole at the origin, so that I've chosen a closed contour that circumvents all the branch points.
I would appreciate it if someone could look at my approach in the graphic below, and let me know if I'm on the right track. If I am, I have a follow-up question about the integration around the branch points, and will update the post accordingly. Note that since there is an underlying physical problem, I am only interested in the real part of the Bromwich integral.
Many thanks in advance.
