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. The paper is at this page, and 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 $L^{-1}[\tilde I(s)]$. I think 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 anyway. 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.

Could anyone please help point out what he was doing? Thanks very much!

By the way, the author also said the inverse Laplace transform could also be evaluated using Laplace transform tables. And the result involved a convolution of the modified Bessel function of the first order. I still haven't worked it out either...