Evaluation of the Riemann-Mellin integral

Abstract

A method for evaluating the Riemann-Mellin integral $$ f(t) = \frac{1} {{2\pi i}}\int\limits_{c - i\infty }^{c + i\infty } {e^{zt} F(z)dz,c > 0,} $$ which determines the inverse Laplace transform, is considered; the method consists in reducing the integral to the form I = ∝ −∞ ∞ g(u) by means of a suitable deformation of the contour of integration and applying the trapezoidal quadrature formulas with an infinite number of nodes (I h = hΣ k=−∞ ∞ g(kh)) or with a finite number 2N + 1 of nodes (I h, N = hΣ k = −N N g(kh)). For parabolic and hyperbolic contours of integration, procedures for choosing the step size h in numerical integration and the summation limits ±N for truncating the infinite sum in the trapezoidal formula, which depend on the arrangement of the singular points of the image, are suggested. Errors are estimated, and their asymptotic behavior with increasing N is described.