Chapman and Hall/CRC
256 pages | 11 B/W Illus.
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.
"…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
Expansions and approximations
Formal and actual solutions
Review of Some Basic Tools
The Phragmén–Lindelöf theorem
Laplace and inverse Laplace transforms
Asymptotics of integrals: first results
Laplace, stationary phase, saddle point methods, and Watson’s lemma
The Laplace method
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
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
Focusing spaces and algebras
Example: Borel summation of the formal solutions to (4.54)
Normalization procedures: an example
Further assumptions and normalization
Overview of results
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
Other Classes of Problems
Other Important Tools and Developments
Resurgence, bridge equations, alien calculus, moulds