Int Jr. of Mathematics Sciences & Applications Vol. 2, No. 2, May 2012 Copyright  Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com 673 BROMWICH INVERSIONI INTEGRAL IN CONTEXT TO NON- MOMENT LAPLACE TRANSFORM Dr. Shobha Lal Associate Professor of Mathematics Jayoti Vidyapeeth Women's University, Jaipur, Rajasthan AND Mr. Rajesh Saxena Research Scholar Jayoti Vidyapeeth Women's University, Jaipur, Rajasthan ABSTRACT We introduce and investigate a framework for constructing algorithms to numerically invert Laplace transforms. Given a Laplace transform f ˆ of a complex-valued function of a nonnegative real-variable, f, the function f is approximated by a nite linear combination of the transform values; i.e., we use the inversion formula where the weights  k and nodes  k are complex numbers, which depend on n, but do not depend on the transform f ˆ or the time argument t. Many different algorithms can be put into this framework, because it remains to specify the weights and nodes. We examine three one-dimensional inversion routines in this framework: the Gaver-Stehfest algorithm, a version of the Fourier-series method with Euler summation, and a version of the Talbot algorithm, which is based on deforming the contour in the Bromwich inversion integral. We show that these three building blocks can be combined to produce different algorithms for numerically inverting two-dimensional Laplace transforms, again all depending on the single parameter n. We show that it can be advantageous to use different one-dimensional algorithms in the inner and outer loops. Keywords : Laplace transforms; numerical transform inversion; Fourier-series method; Talbot's method; Gaver-Stehfest algorithm; multi-precision computing; multidimensional Laplace transforms; multidimensional transform inversion. 1. Introduction In recent years, numerical transform inversion has become recognized as an important technique in operations research, notably for calculating probability distributions in stochastic models. There have now been many applications; e.g., see the survey by Abate et al. (1999) and the textbook treatment by Kao (1997). Over the years, many different algorithms have been proposed for numerically inverting Laplace transforms; e.g., see the surveys in Abate and Whitt (1992) and Chapter 19 of Davies (2002), the extensive bibliography of Valko and Vojta (2001) and the numerical comparisons by Davies and Martin (1979), Narayanan and Beskos (1982) and Duffy (1993). In contrast to the usual approach, in this paper we do not focus on a particular procedure, but instead introduce and investigate a framework that can encompass a wide range of procedures. The flexible framework opens the way to further work, e.g., performing optimization in order to choose the best set of parameters and thus the best procedure, by various criteria, for various classes of functions.