Laplace transform inversion and Padé-type approximants
A method for the numerical inversion of the Laplace transform of a function f(z) is to approximate it by rational functions \(f_ m(z)\), and then to use the inverse transforms \(F_ m(t)\) of \(f_ m(z)\) as approximations of the inverse transform F(t) of f(z). As in Tricomi's method the author defines \(f_ m(z)\) as a partial sum of a series expansion, which is also a Padé-type approximant to f with one pole. Then \(F_ m(t)\) is the partial sum of the expansion of F(t) in terms of Laguerre polynomials. The author proves convergence results in the mean-squares and uniform norms if f(z) is analytic in the half-plane Re \(z\geq 0\), and \((z+\lambda)\) f(z) analytic at infinity. The computations are very sensitive to the choice of \(\lambda\), especially when the function has singularities which are not poles. She uses the study for the choice of the pole of \(f_ m\) to define a best Padé-type approximant with one pole. This permits the use of the method of inversion by Laguerre polynomials. Favorable numerical results for functions having essential singularities are reported.
numerical inversion of the Laplace transform
scientific article; zbMATH DE number 4033018