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Full Description
This three-volume work treats divergent series in one variable, especially those arising as solutions to complex ordinary differential or difference equations, and methods for extracting their analytic information.It provides a systematic construction, illustrated with examples, of the various theories of summability and the theory of resurgence developed since the 1980s. The Stokes phenomenon, for both linear and non-linear equations, plays an underlying and unifying role throughout the volumes.Applications presented include resurgent analyses of the First Painleve equation and of the tangent-to-identity germs of diffeomorphisms of C, and links to differential Galois theory and the Riemann-Hilbert problem for linear differential equations.The volumes are aimed at graduate students, mathematicians in general, and theoretical physicists who are interested in the theories of monodromy, summability, and resurgence, as well as the current problems in the field.Although the three volumes are closely related, they have been organized to be read independently. The prerequisites are advanced calculus, especially holomorphic functions in one complex variable, and differential algebra. Moreover; the various themes are presented thoroughly step-by-step so as to be accessible to first-year graduate students in mathematics.This three-volume treatise should become a reference on summability and resurgence.
Contents
Preface.-Preface to the three volumes.- Part I:Monodromy in Linear Differential Equations.- 1 analytic continuation and monodromy.- Differential Galois Theory.- Inverse Problems.- The Riemann-Hilbert problem.- Part II: Introduction to 1-Summability and Resurgence.- 5 Borel-Laplace Summation.- Resurgent Functions and Alien Calculus.- the Resurgent Viewpoint on Holomorphic Tangent-to-Identity Germs.- Acknowledgements.- Index.Avant-propos.- Preface to the three volumes.- Introduction to this volume.- 1 Asymptotic Expansions in the Complex Domain.- 2 Sheaves and Cech cohomology.- 3 Linear Ordinary Differential Equations.- 4 Irregularity and Gevrey Index Theorems.- 5 Four Equivalent Approaches to k-Summability.- 6 Tangent-to-Identity Diffeomorphisms.- 7 Six Equivalent Approaches to Multisummability.- Exercises.- Solutions to Exercises.- Index.- Glossary of Notations.- References.Avant-Propos.- Preface to the three volumes.- Preface to this volume.- Some elements about ordinary differential equations.- The first Painleve equation.- Tritruncated solutions for the first Painleve equation.- A step beyond Borel-Laplace summability.- Transseries and formal integral for the first Painleve equation.- Truncated solutions for the first Painleve equation.- Supplements to resurgence theory.- Resurgent structure for the first Painleve equation.- Index.
