On the summation of the Laguerre series by the Euler-Knopp method in the problem of inverting the Laplace transform

Abstract

A method for inverting the Laplace transform based on expanding the original in Laguerre polynomials as $$ f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt)} $$ is suggested. The representation of the Laguerre series by a linear-fractional mapping is reduced to a power series of the form \( \sum\nolimits_{k = 0}^\infty {a_k z^k } \), which is summed by the well-known Euler-Knopp method. The summation parameter is chosen in the complex plane so that the new expansion $$ f(t) = \exp \left( {\frac{{bpt}} {{p - 1}}} \right)\sum\limits_{k = 0}^\infty {\frac{{A_k (p)}} {{(1 - p)^{k + 1} }}L_k \left( {\frac{{bpt}} {{1 - p}}} \right)} $$ of the original corresponding to the Euler-Knopp transformation converge at a maximum rate. On the basis of geometric representations, the influence of the requirement that the Euler-Knopp transformation must be regular on the choice of the summation parameter is discussed. Numerical experiments are performed, which demonstrate the high efficiency of the method of choosing a complex parameter suggested in this paper.