Numer. Math. 54, 193-200 (1988) Numerische Mathematik 9 Springer-Verlag 1988 More on the Weeks Method for the Numerical Inversion of the Laplace Transform G. Giunta, G. Laccetti, and M.R. Rizzardi Dipartimento di Matematica e Applicazioni, Universit/l di Napoli, Via Mezzocannone 16, 1-80134 Napoli, Italy Summary. Most of the numerical methods for the inversion of the Laplace Transform require the values of several incidental parameters. Generally, these parameters are related to the properties of the algorithm and to the analytical properties of the Laplace Transform function F(s). One of the most promising inversion methods, the Weeks methods, com- putes the inverse function f(t) as a series expansion of Laguerre functions involving two parameters, usually denoted by a and b. In this paper we characterize the optimal choice bopt of b, which maximizes the rate of conver- gence of the series, in terms of the location of the singularities of F(s). Subject Classifications: AMS(MOS) 65R10, 44A10; CR G 1.9. 1. Introduction The problem of the inversion of a Laplace Transform consists in expressing a function f(t) in terms of its Laplace Transform F(s). The Weeks method [4, 6] is based on the following series representation, established first by Tricomi ES]: f(t) =e~' L ake-bt/2 Lk( bt)' (1.1) k=O where the coefficients ak are the Taylor coefficients of eP(z;a'b)=-lb-z F 1 -z +a-b~2 ak zk, ]z[<R~(a,b), (1.2) a>a o, b>O are parameters, and Lk(x ) is the k-th Laguerre polynomial. As usual, a o denotes the abscissa of convergence of the Laplace Transform F(s), which can be defined as the maximum of the real parts of the singularities of F(s).