An Extended Padé Procedure for Constructing Global Approximations from Asymptotic Expansions: An Explication with Examples

The problem of generating approximations to functions, specified by asymptotic expansions about one or more points, is considered. The development is limited to expansions of the power series type or expansions which assume a power series structure following a suitable transformation. Procedures are described for incorporating a variety of quantitative and qualitative information into rational fraction approximations, subject to the constraint that the approximants are, at the same time, asymptotically equivalent to the given expansions. These procedures are then illustrated by a set of five examples, taken from the literature of fluid mechanics, which either emphasize a particular guideline or are of some practical importance. In all cases the approximations presented are superior to those that have previously been published.