Asymptotics and Borel Summability

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. Les mer
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Om boka

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.

Fakta

Innholdsfortegnelse

Introduction


Expansions and approximations


Formal and actual solutions


Review of Some Basic Tools


The Phragmen-Lindeloef 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

Om forfatteren

Ohio State University, Columbus, USA University of Newcastle upon Tyne, UK Centre National de La Recherche Scientifique/College of Fran Texas A & M University