Asymptotics and Borel Summability  book cover
1st Edition

Asymptotics and Borel Summability

ISBN 9781420070316
Published December 4, 2008 by Chapman and Hall/CRC
256 Pages 11 B/W Illustrations

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Book Description

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.

Table of Contents


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


Singular perturbations

WKB on a PDE

Analyzable Functions and Transseries

Analytic function theory as a toy model of the theory of analyzable functions


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: 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


Other Important Tools and Developments

Resurgence, bridge equations, alien calculus, moulds





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"…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