Uniformization of Asymptotic Expansions

We develop a general technique for uniformizing asymptotic expansions. A basic formula for uniformizing counterterms is constructed by nesting an increasing number of extensions. Two successive extensions are shown to contain the counterterms for both secular terms and for singular perturbation terms. Simple examples are used to illustrate the method. A general constructive procedure is outlined to determine the counterterms from the nonuniformities arising in the direct perturbation expansion. A relationship is established between the secular perturbation counterterms and the singular perturbation counterterms. Finally, a brief review is given of the main results obtained so far and of the open problems.