1st Edition

Asymptotics and Borel Summability

By Ovidiu Costin Copyright 2008
    256 Pages 11 B/W Illustrations
    by Chapman & Hall

    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.

    Introduction

    Expansions and approximations

    Formal and actual solutions

    Review of Some Basic Tools

    The Phragmén–Lindelöf 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 ξ 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

    Biography

    Ovidiu Costin

    "…Until the welcome publication of Ovidiu Costin’s textbook on the subject, this more modern approach to asymptotic analysis was only accessible via original research papers and a few technical lecture note publications. …the author’s perspective provides some key new insights not found in [traditional] books. …One of the great strengths of this book is its use of many examples to illustrate general theory. … A good first course on the subject of transseries and Borel summation could be designed around these examples. …The author maintains a comprehensive list of corrections organized by page number on the web … In summary, as one of a small number of experts in the subject of transseries and Borel summation, Ovidiu Costin has written a book that will be a fundamental reference to researchers and students interested in going beyond the standard classical methods of asymptotic analysis."
    Journal of Approximation Theory, 2010

    "This important new book is about asymptotics beyond all orders, i.e., recovering actual solutions from formal expansions. The book goes far beyond the logarithmico-exponentials of Hardy and the Borel–Ritt theory of Wasow by utilizing recent work of Ecalle and Costin, among others. … This unique monograph should stimulate a broad new effort to demystify the use of asymptotic series."
    SIAM Review, Volume 51, Issue 3, 2009