The inverse Laplace transform of a function (probably numerically)

I originally asked this question on MathOverflow but it was regarded as not being "research level". I repost the question here (hopefully it falls within forum's category this time) and will really appreciate if someone can shine light on it.

I need to do an inverse Laplace transform of a function with essential singularities for a specific problem. I find it is very similar to an equation J. Noolandi worked out in one of his papers in 1977 (Phys. Rev. B 16, 4466 (1977)). I have taken a snapshot of the Appendix page where the math problem and the solution were given (view it here). But I don't quite get some part of the solving process which I need help on deciphering.

If I can understand what he was doing, my specific problem can be solved. So now let's take the problem in the Appendix of Noolandi's paper for example. The function is

$\displaystyle \tilde I(s) = \frac{1-\exp(-a(s)t_0)}{a(s)}$

where $\displaystyle a(s)t_0 = \Sigma_{i=1}^n \frac{s\omega_i}{s+r_i} t_0 = \Sigma_{i=1}^n \frac{sM_i}{s+r_i}$. "$s$" is the variable. $\omega_i$, $r_i$, $t_0$, and $M_i$ are all real numbers greater than zero.

The question is how to get inverse Laplace transform $L^{-1}[\tilde I(s)]$. It looks like the author used Residue Theorem to evaluate the contour integral $\int_C \tilde I(s) \exp(st) ds$, but the hard part is the singularities are essential singularities, thus the residues can not be got conventionally.

However, it seems that the author constructed a contour and solved it numerically. I don't get Equation A5-A8 in the Appendix, and don't understand how to determine the parameter $R$ and $R_1$ he introduced in (probably something related to numerics).