Asymptotics And Borel Summability

Author : Ovidiu Costin
Publisher : CRC Press
Page : 256 pages
File Size : 42,6 Mb
Release : 2008-12-04
Category : Mathematics
ISBN : 1420070320

Get Book

Asymptotics and Borel Summability by Ovidiu Costin Pdf

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.

Author : Ovidiu Costin,Frédéric Fauvet,Frédéric Menous,David Sauzin
Publisher : Springer Science & Business Media
Page : 274 pages
File Size : 43,7 Mb
Release : 2012-02-21
Category : Mathematics
ISBN : 9788876423772

Get Book

Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation by Ovidiu Costin,Frédéric Fauvet,Frédéric Menous,David Sauzin Pdf

These are the proceedings of a one-week international conference centered on asymptotic analysis and its applications. They contain major contributions dealing with: mathematical physics: PT symmetry, perturbative quantum field theory, WKB analysis, local dynamics: parabolic systems, small denominator questions, new aspects in mould calculus, with related combinatorial Hopf algebras and application to multizeta values, a new family of resurgent functions related to knot theory.

Author : Victor Kowalenko
Publisher : Unknown
Page : 262 pages
File Size : 54,9 Mb
Release : 2018-02-22
Category : Mathematics
ISBN : 1608050971

Get Book

The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation by Victor Kowalenko Pdf

The Stokes phenomenon refers to the emergence of jump discontinuities in asymptotic expansions at specific rays in the complex plane. This book presents a radical theory for the phenomenon by introducing the concept of regularization. Two methods of regularization, Borel summation and Mellin-Barnes regularization, are used to derive general expressions for the regularized values of asymptotic expansions throughout the complex plane. Though different, both yield identical values, which, where possible, agree with the original functions. Consequently, asymptotics has been elevated to a true discipline yielding precise solutions. All researchers, who seek asymptotic solutions to problems, will find this a most valuable book.

Author : Harvey Segur,Saleh Tanveer,Herbert J. Levine
Publisher : Springer Science & Business Media
Page : 388 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781475704358

Get Book

Asymptotics beyond All Orders by Harvey Segur,Saleh Tanveer,Herbert J. Levine Pdf

An asymptotic expansion is a series that provides a sequence of increasingly accurate approximations to a function in a particular limit. The formal definition, given by Poincare (1886, Acta Math. 8:295), is as follows. Given a function,

Author : Frédéric Fauvet,Dominique Manchon,Stefano Marmi,David Sauzin
Publisher : Springer
Page : 384 pages
File Size : 49,9 Mb
Release : 2017-11-17
Category : Science
ISBN : 9788876426131

Get Book

Resurgence, Physics and Numbers by Frédéric Fauvet,Dominique Manchon,Stefano Marmi,David Sauzin Pdf

This book is issued from a conference around resurgent functions in Physics and multiple zetavalues, which was held at the Centro di Ricerca Matematica Ennio de Giorgi in Pisa, on May 18-22, 2015. This meeting originally stemmed from the impressive upsurge of interest for Jean Ecalle's alien calculus in Physics, in the last years – a trend that has considerably developed since then. The volume contains both original research papers and surveys, by leading experts in the field, reflecting the themes that were tackled at this event: Stokes phenomenon and resurgence, in various mathematical and physical contexts but also related constructions in algebraic combinatorics and results concerning numbers, specifically multiple zetavalues.

Author : Eric Delabaere
Publisher : Springer
Page : 230 pages
File Size : 54,9 Mb
Release : 2016-06-28
Category : Mathematics
ISBN : 9783319290003

Get Book

Divergent Series, Summability and Resurgence III by Eric Delabaere Pdf

The aim of this volume is two-fold. First, to show how the resurgent methods introduced in volume 1 can be applied efficiently in a non-linear setting; to this end further properties of the resurgence theory must be developed. Second, to analyze the fundamental example of the First Painlevé equation. The resurgent analysis of singularities is pushed all the way up to the so-called “bridge equation”, which concentrates all information about the non-linear Stokes phenomenon at infinity of the First Painlevé equation. The third in a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists who are interested in divergent power series and related problems, such as the Stokes phenomenon. The prerequisites are a working knowledge of complex analysis at the first-year graduate level and of the theory of resurgence, as presented in volume 1.

Author : Norman Bleistein,Richard A. Handelsman
Publisher : Courier Corporation
Page : 453 pages
File Size : 41,5 Mb
Release : 1986-01-01
Category : Mathematics
ISBN : 9780486650821

Get Book

Asymptotic Expansions of Integrals by Norman Bleistein,Richard A. Handelsman Pdf

Excellent introductory text, written by two experts, presents a coherent and systematic view of principles and methods. Topics include integration by parts, Watson's lemma, LaPlace's method, stationary phase, and steepest descents. Additional subjects include the Mellin transform method and less elementary aspects of the method of steepest descents. 1975 edition.

Author : Galina Filipuk,Alberto Lastra,Slawomir Michalik
Publisher : World Scientific
Page : 400 pages
File Size : 43,5 Mb
Release : 2022-03-03
Category : Mathematics
ISBN : 9781800611375

Get Book

Formal And Analytic Solutions Of Differential Equations by Galina Filipuk,Alberto Lastra,Slawomir Michalik Pdf

The book provides the reader with an overview of the actual state of research in ordinary and partial differential equations in the complex domain. Topics include summability and asymptotic study of both ordinary and partial differential equations, and also q-difference and differential-difference equations. This book will be of interest to researchers and students who wish to expand their knowledge of these fields.With the latest results and research developments and contributions from experts in their field, Formal and Analytic Solutions of Differential Equations provides a valuable contribution to methods, techniques, different mathematical tools, and study calculations.

Author : Athanassios S. Fokas,Alexander R. Its,Andrei A. Kapaev,Victor Yu. Novokshenov
Publisher : American Mathematical Society
Page : 570 pages
File Size : 41,9 Mb
Release : 2023-11-20
Category : Mathematics
ISBN : 9781470475567

Get Book

Painlevé Transcendents by Athanassios S. Fokas,Alexander R. Its,Andrei A. Kapaev,Victor Yu. Novokshenov Pdf

At the turn of the twentieth century, the French mathematician Paul Painlevé and his students classified second order nonlinear ordinary differential equations with the property that the location of possible branch points and essential singularities of their solutions does not depend on initial conditions. It turned out that there are only six such equations (up to natural equivalence), which later became known as Painlevé I–VI. Although these equations were initially obtained answering a strictly mathematical question, they appeared later in an astonishing (and growing) range of applications, including, e.g., statistical physics, fluid mechanics, random matrices, and orthogonal polynomials. Actually, it is now becoming clear that the Painlevé transcendents (i.e., the solutions of the Painlevé equations) play the same role in nonlinear mathematical physics that the classical special functions, such as Airy and Bessel functions, play in linear physics. The explicit formulas relating the asymptotic behaviour of the classical special functions at different critical points play a crucial role in the applications of these functions. It is shown in this book that even though the six Painlevé equations are nonlinear, it is still possible, using a new technique called the Riemann-Hilbert formalism, to obtain analogous explicit formulas for the Painlevé transcendents. This striking fact, apparently unknown to Painlevé and his contemporaries, is the key ingredient for the remarkable applicability of these “nonlinear special functions”. The book describes in detail the Riemann-Hilbert method and emphasizes its close connection to classical monodromy theory of linear equations as well as to modern theory of integrable systems. In addition, the book contains an ample collection of material concerning the asymptotics of the Painlevé functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Painlevé equations and related areas.

Author : Terence Tao
Publisher : American Mathematical Soc.
Page : 206 pages
File Size : 55,8 Mb
Release : 2021-09-03
Category : Education
ISBN : 9781470466404

Get Book

An Introduction to Measure Theory by Terence Tao Pdf

This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

Author : Ricardo Estrada,Ram P. Kanwal
Publisher : Springer Science & Business Media
Page : 467 pages
File Size : 50,9 Mb
Release : 2012-09-08
Category : Mathematics
ISBN : 9780817681302

Get Book

A Distributional Approach to Asymptotics by Ricardo Estrada,Ram P. Kanwal Pdf

"...The authors of this remarkable book are among the very few who have faced up to the challenge of explaining what an asymptotic expansion is, and of systematizing the handling of asymptotic series. The idea of using distributions is an original one, and we recommend that you read the book...[it] should be on your bookshelf if you are at all interested in knowing what an asymptotic series is." -"The Bulletin of Mathematics Books" (Review of the 1st edition) ** "...The book is a valuable one, one that many applied mathematicians may want to buy. The authors are undeniably experts in their field...most of the material has appeared in no other book." -"SIAM News" (Review of the 1st edition) This book is a modern introduction to asymptotic analysis intended not only for mathematicians, but for physicists, engineers, and graduate students as well. Written by two of the leading experts in the field, the text provides readers with a firm grasp of mathematical theory, and at the same time demonstrates applications in areas such as differential equations, quantum mechanics, noncommutative geometry, and number theory. Key features of this significantly expanded and revised second edition: * addition of a new chapter and many new sections * wide range of topics covered, including the Ces.ro behavior of distributions and their connections to asymptotic analysis, the study of time-domain asymptotics, and the use of series of Dirac delta functions to solve boundary value problems * novel approach detailing the interplay between underlying theories of asymptotic analysis and generalized functions * extensive examples and exercises at the end of each chapter * comprehensive bibliography and index This work is an excellent tool for the classroom and an invaluable self-study resource that will stimulate application of asymptotic

Author : Igor V. Andrianov,Jan Awrejcewicz
Publisher : CRC Press
Page : 265 pages
File Size : 51,6 Mb
Release : 2024-05-16
Category : Mathematics
ISBN : 9781040032718

Get Book

Asymptotic Methods for Engineers by Igor V. Andrianov,Jan Awrejcewicz Pdf

Asymptotic Methods for Engineers is based on the authors’ many years of practical experience in the application of asymptotic methods to solve engineering problems. This book is devoted to modern asymptotic methods (AM), which is widely used in engineering, applied sciences, physics, and applied mathematics. Avoiding complex formal calculations and justifications, the book’s main goal is to describe the main ideas and algorithms. Moreover, not only is there a presentation of the main AM, but there is also a focus on demonstrating their unity and inseparable connection with the methods of summation and asymptotic interpolation. The book will be useful for students and researchers from applied mathematics and physics and of interest to doctoral and graduate students, university and industry professors from various branches of engineering (mechanical, civil, electro-mechanical, etc.).

Author : Michèle Loday-Richaud
Publisher : Springer
Page : 272 pages
File Size : 54,7 Mb
Release : 2016-06-28
Category : Mathematics
ISBN : 9783319290751

Get Book

Divergent Series, Summability and Resurgence II by Michèle Loday-Richaud Pdf

Addressing the question how to “sum” a power series in one variable when it diverges, that is, how to attach to it analytic functions, the volume gives answers by presenting and comparing the various theories of k-summability and multisummability. These theories apply in particular to all solutions of ordinary differential equations. The volume includes applications, examples and revisits, from a cohomological point of view, the group of tangent-to-identity germs of diffeomorphisms of C studied in volume 1. With a view to applying the theories to solutions of differential equations, a detailed survey of linear ordinary differential equations is provided, which includes Gevrey asymptotic expansions, Newton polygons, index theorems and Sibuya’s proof of the meromorphic classification theorem that characterizes the Stokes phenomenon for linear differential equations. This volume is the second in a series of three, entitled Divergent Series, Summability and Resurgence. It is aimed at graduate students and researchers in mathematics and theoretical physics who are interested in divergent series, Although closely related to the other two volumes, it can be read independently.

Author : Domingos H. U. Marchetti,Walter F. Wreszinski
Publisher : World Scientific
Page : 362 pages
File Size : 43,8 Mb
Release : 2013
Category : Science
ISBN : 9789814383806

Get Book

Asymptotic Time Decay in Quantum Physics by Domingos H. U. Marchetti,Walter F. Wreszinski Pdf

Time decays form the basis of a multitude of important and interesting phenomena in quantum physics that range from spectral properties, resonances, return and approach to equilibrium, to quantum mixing, dynamical stability preperties and irreversibility and the "arrow of time." This monograph is devoted to a clear and precise, yet pedagogical account of the associated concepts and methods.

Author : Boris Yu Sternin,Victor E. Shatalov
Publisher : Unknown
Page : 0 pages
File Size : 52,6 Mb
Release : 1996
Category : Electronic
ISBN : OCLC:637715321

Get Book

Borel-Laplace Transform and Asymptotic Theory by Boris Yu Sternin,Victor E. Shatalov Pdf