Costin Asymptotics and Borel Summability

Erscheinungsjahr 2010
ISBN: 978-1-4200-7032-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 256 Seiten

Reihe: Monographs and Surveys in Pure and Applied Mathematics

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.

Zielgruppe

Pure and applied mathematicians, as well as physicists and engineers and students in these fields

Autoren/Hrsg.

Costin, Ovidiu

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein

Inhaltsverzeichnis

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 ? 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