Divergent series just got more convergent

Abstract

Asymptotic analysis is the field of mathematics that deals with an approximation of solutions to differential equations or of functions in integral representations by means of divergent infinite series called asymptotic series. In this talk I will discuss how divergent series naturally arises as perturbative solutions to differential equations of physics; and, hence, show the relevance of asymptotic analysis in shedding significant insight into the inner workings for the physical universe. From there I will discuss the limitations of the current theory of asymptotics in dealing with non-perturbative effects. I will then present the recent contribution of my research group in dealing with the problem of extending the reach of the standard generalized Poincare asymptotic expansions into the non-asymptotic or non-perturbative regime. My discussion will focus on the discovery of my group that the reach of the standard asymptotic series can be extended beyond the asymptotic regime by means of non-asymptotic scales. I will conclude by discussing the ramifications of introducing non-asymptotic scales in asymptotic analysis, in particular the need to reconsider the foundations of the theory of asymptotic analysis itself.