Incorporating substantial developments from the last thirty years into one resource, Asymptotics and Borel Summability provides a self-contained introduction to asymptotic analysis with special emphasis on topics not covered in traditional asymptotics books. The author explains basic ideas, concepts, and methods of generalized Borel summability, transseries, and exponential asymptotics. He provides complete mathematical rigor while supplementing it with heuristic material and examples, so that some proofs may be omitted by applications-oriented readers. To give a sense of how new methods are used in a systematic way, the book analyzes in detail general nonlinear ordinary differential equations (ODEs) near a generic irregular singular point. It enables readers to master basic techniques, supplying a firm foundation for further study at more advanced levels. The book also examines difference equations, partial differential equations (PDEs), and other types of problems.
Chronicling the progress made in recent decades, this book shows how Borel summability can recover exact solutions from formal expansions, analyze singular behavior, and vastly improve accuracy in asymptotic approximations.
Ohio State University, Columbus, USA University of Newcastle upon Tyne, UK Centre National de La Recherche Scientifique/College of Fran Texas A & M University
Introduction Expansions and approximations Formal and actual solutions Review of Some Basic Tools The Phragmen-Lindelof theorem Laplace and inverse Laplace transforms Classical Asymptotics Asymptotics of integrals: first results Laplace, stationary phase, saddle point methods, and Watson's lemma The Laplace method Watson's lemma Oscillatory integrals and the stationary phase method Steepest descent method application: asymptotics of Taylor coefficients of analytic functions Banach spaces and the contractive mapping principle Examples Singular perturbations WKB on a PDE Analyzable Functions and Transseries Analytic function theory as a toy model of the theory of analyzable functions Transseries Solving equations in terms of Laplace transforms Borel transform, Borel summation Gevrey classes, least term truncation, and Borel summation Borel summation as analytic continuation Notes on Borel summation Borel transform of the solutions of an example ODE Appendix: rigorous construction of transseries Borel Summability in Differential Equations Convolutions revisited Focusing spaces and algebras Example: Borel summation of the formal solutions to (4.54) General setting Normalization procedures: an example Further assumptions and normalization Overview of results Further notation Analytic properties of Yk and resurgence Outline of the proofs Appendix Appendix: the C*-algebra of staircase distributions, D'm,v Asymptotic and Transasymptotic Matching; Formation of Singularities Transseries reexpansion and singularities: Abel's equation Determining the xi reexpansion in practice Conditions for formation of singularities Abel's equation, continued General case Further examples Other Classes of Problems Difference equations PDEs Other Important Tools and Developments Resurgence, bridge equations, alien calculus, moulds Multisummability Hyperasymptotics References Index
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