I am seeking a references that provide a rigorous treatment of the inverse Laplace transform (Bromwich integrals), and how to compute them (beyond using tabled solutions - they don't cover my needs, so I'm trying to learn to do it via complex analysis)? Textbooks or papers would be great. I would prefer something with lots of examples worked out if possible, my experience with complex integration is a bit shallow.
To clarify, I'd like to understand how to handle integrals like this:
$$ \mathscr{L}^{-1}_s\lbrace F(s)\rbrace=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} e^{st}F(s) ds $$
for a real number $$\gamma > Re(s)$$ for each singularity s of F.
Should I just be looking at the calculus of residues, or is there something special about integrals of this sort? It appears a bit different than what I've seen in my complex analysis text - there I would always see integrals of this sort:
$$ \int_L f(s)ds $$
But L would always be a closed rectifiable curve, while the inverse Laplace transform appears to be over a vertical line in the complex plane, which I'm not sure how to interpret. Perhaps it's just a notational misunderstanding, and I'm meant to construct a suitable contour expanding on that vertical line?
Any direction would be greatly appreciated!
