Quadrature formulas for the Laplace and Mellin transforms
Provides new numerical methods for computing Laplace and Mellin transforms, which are important tools in applied mathematics and engineering.
The paper develops convergent quadrature formulas for the Laplace and Mellin transforms and their inverses using Hermite polynomials, achieving convergence for rapidly decaying and singular functions.
A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the continuous Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula for the two-sided Laplace transform if the function to be transformed falls off rapidly to zero and satisfy certain conditions of integrability, achieving convergence also for singular functions. The inversion formula becomes a quadrature formula for the Bromwich integral. This procedure also yields a quadrature formula for the Mellin transform and its corresponding inversion formula that can be generalized straightforwardly for functions of several variables.