BIT 9 (1969), 351--361 NEW QUADRATURE FORMULAS FOR THE NUMERICAL INVERSION OF THE LAPLACE TRANSFORM ROBERT PIESSENS Abstract. New quadrature formulas for the evaluation of the Bromwich integral, arising in the inversion of the Laplace transform are discussed. They are obtained by optimal addition of abscissas to Gaussian quadrature formulas. A table of abscissas and weights is given. I. Introduction. The main difficulty in applying Laplace transform techniques is the determination of the original function f(t) from its transform CO (1) F(p) = I e-~t f ( t ) dt . o In many cases numerical methods must be used. The computation of f(t) from values of F(p) on the real axis is not well-posed [1], [2], [3], [4], [5]. Thus, if high accuracy is required, the calculation must be carried out in multiple precision, or methods must be used which determine the original function from values of the transform in the complex plane. The best known methods of this type are given in [6], [7], [8], and [9]. A very simple method is the numerical integration of the Bromwich inversion formula 1 ~ t (2) f(t) = ~ ! e ~ F(p)dp where L is defined as the line {p: R(p)=c} in the complex plane and where c is chosen so that L lies to the right of all the singularities of _F(p), but is otherwise arbitrary. Substituting Received l~Iay 22, 1969. and pt=u