Asymptotic Expansions in the Complex Domain

Abstract

We introduce asymptotic expansions on sectors with vertex 0 in the complex plane, both in the Poincaré and the Gevrey sense. Gevrey asymptotics are an essential link between divergent series and their sums. We prove basic properties of asymptotic expansions and pay special attention to the case of flat Gevrey-asymptotic functions, that is, asymptotic to 0 in the Gevrey sense. We look at the action of a finite extension of the variable $x = t^r$ and of rank reduction. We prove the Borel-Ritt theorem which extends to the case of a complex variable the classical Borel theorem for a real variable. We end the chapter with the Cauchy-Heine theorem which links (non zero) asymptotic expansions to flat functions, considering both the case of Poincaré and Gevrey asymptotics.