Robert Piessens

Gaussian quadrature formulas for the numerical integration of Bromwich's integral and the inversion of the laplace transform

SummaryAn approximate formula for the inversion of the Laplace transformF(p) is studied. The formula is exact wheneverF(p) is a linear combination ofp−s+k,k=0, 1, 2, ..., 2N−1, withs an arbitrary positive real number. The formula is derived from a gaussian integration formula for Bromwichs inversion integral.A numerical example is given as illustration of the use of the approximate inversion formula.

New quadrature formulas for the numerical inversion of the Laplace transform

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.

Numerical inversion of the Laplace transform

An extension of Bellmans method for the numerical inversion of the Laplace transform is discussed. This extension is theoretically equivalent to the method of Lanczos. Tables of coefficients are given which facilitate the inversion of the Laplace transform with the aid of a desk computer.

ABACUS: a powerful tool in a mathematical sciences laboratory

This paper describes a highly interactive computer program, ABACUS, which exploits the potential of microcomputers in a mathematical sciences laboratory. In contrast to most educational software which demonstrates very specific points, ABACUS can be used for a variety of applications: it allows one to plot one or more functions on the same set of axes, to differentiate analytically, to compute a definite integral, to search the zeros of a function, and to solve systems of differential equations and to generate Taylor series expansions. Some applications demonstrate the usefulness of ABACUS.

Computation of Fourier and Laplace transforms of singular functions using modified moments

Abstract In this paper a recurrence formula for the computation of M k = ∫ −1 +1 (1 − x) α (1 + x) β exp [−a/(1+x)]T k (x) d x is presented. The numerical stability is discussed. The starting values are confluent hypergeometric functions which can be evaluated using Lukes results on Chebyshev series expansions and Pade approximations of hypergeometric functions. Applications of this recurrence relation are the evaluation of the Fourier transform of singular functions by modified Clenshaw-Curtis integration, the construction of Gaussian quadrature formulae for Fourier integrals and the numerical inversion of the Laplace transform.