Thanks for your reply. (1) Sorry about the ambiguity. What I was trying to say was I needed to understand how to do the inverse Laplace transform of function like $\tilde I(s)$ in that paper. Once I understood that, my real question would be solved consequently. (2) The answer doesn't have to be analytic. I think Noolandi gave the guideline for doing the transform numerically. But the problem is I don't understand his approach on the paper. (3) Yes. But expanding the function in Laurent series and filtering out the z^{-1} term is not easy. I am hoping for a more clear.