Divergent Series, Summability and Resurgence II: Simple and Multiple Summability

Description

Addressing the question how to “sum” a power series in one variable when it diverges, that is, how to attach to it analytic functions, the volume gives answers by presenting and comparing the various theories of k-summability and multisummability. These theories apply in particular to all solutions of ordinary differential equations. The volume includes applications, examples and revisits, from a cohomological point of view, the group of tangent-to-identity germs of diffeomorphisms of C studied in volume 1. With a view to applying the theories to solutions of differential equations, a detailed survey of linear ordinary differential equations is provided, which includes Gevrey asymptotic expansions, Newton polygons, index theorems and Sibuya’s proof of the meromorphic classification theorem that characterizes the Stokes phenomenon for linear differential equations. This volume is the second in a series of three, entitled Divergent Series, Summability and Resurgence. It is aimed at graduate students and researchers in mathematics and theoretical physics who are interested in divergent series, Although closely related to the other two volumes, it can be read independently. Front Matter....Pages i-xxiii Asymptotic Expansions in the Complex Domain....Pages 1-31 Sheaves and Čech Cohomology with an Insight into Asymptotics....Pages 33-62 Linear Ordinary Differential Equations: Basic Facts and Infinitesimal Neighborhoods at an Irregular Singular Point....Pages 63-120 Irregularity and Gevrey Index Theorems for Linear Differential Operators....Pages 121-132 Four Equivalent Approaches to k-Summability....Pages 133-183 Tangent-to-Identity Diffeomorphisms and the Birkhoff Normalization Theorem....Pages 185-195 Six Equivalent Approaches to Multisummability....Pages 197-235 Back Matter....Pages 237-272