A New Method for Generating Power Series Expansions of Functions

Abstract

This paper derives a numerical scheme for determining the coefficients of the power series expansion of a given function about the origin. The Laplace transform of the partial sums of coefficients is developed by knowledge of the given function near the origin where it is assumed to be analytic. This transform is then inverted to any desired accuracy by making use of a new inversion technique. The method obtains any partial sum of coefficients directly, thereby yielding the cumulative distribution of a probability generating function immediately. The computational steps, of which there are very few, have been arranged to facilitate automatic digital computation, and numerical illustrations have been given.