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

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Springer, Jun 28, 2016 - Mathematics - 272 pages
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.

 

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Contents

Chapter 1 Asymptotic Expansions in the Complex Domain
1
Chapter 2 Sheaves and Čech Cohomology with an Insight into Asymptotics
33
Basic Facts and Infinitesimal Neighborhoods at an Irregular Singular Point
63
Chapter 4 Irregularity and Gevrey Index Theorems for Linear Differential Operators
121
Chapter 5 Four Equivalent Approaches to KSummability
133
Chapter 6 TangenttoIdentity Diffeomorphisms and the Birkhoff Normalization Theorem
185
Chapter 7 Six Equivalent Approaches to Multisummability
197
Chapter 8 Exercises
236
Chapter 9 Solutions to Exercises
245
References
264
Glossary of Notations
268
Index
270
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