A Distributional Approach to Asymptotics

Theory and Applications
2. Auflage 2002
ISBN: 978-0-8176-4142-9
Verlag: Birkhäuser Boston

Buch, Englisch, 454 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 1880 g

Reihe: Birkhäuser Advanced Texts Basler Lehrbücher

".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

Zielgruppe

Research

Autoren/Hrsg.

Kanwal, Ram P.

Estrada, Ricardo

Fachgebiete

• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik
• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis
• Naturwissenschaften Physik Quantenphysik
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

1 Basic Results in Asymptotics.- 1.1 Introduction.- 1.2 Order Symbols.- 1.3 Asymptotic Series.- 1.4 Algebraic and Analytic Operations.- 1.5 Existence of Functions with a Given Asymptotic Expansion.- 1.6 Asymptotic Power Series in a Complex Variable.- 1.7 Asymptotic Approximation of Partial Sums.- 1.8 The Euler-Maclaurin Summation Formula.- 1.9 Exercises.- 2 Introduction to the Theory of Distributions.- 2.1 Introduction.- 2.2 The Space of Distributions D?.- 2.3 Algebraic and Analytic Operations.- 2.4 Regularization, Pseudofunction and Hadamard Finite Part.- 2.5 Support and Order.- 2.6 Homogeneous Distributions.- 2.7 Distributional Derivatives of Discontinuous Functions.- 2.8 Tempered Distributions and the Fourier Transform.- 2.9 Distributions of Rapid Decay.- 2.10 Spaces of Distributions Associated with an Asymptotic Sequence.- 2.11 Exercises.- 3 A Distributional Theory for Asymptotic Expansions.- 3.1 Introduction.- 3.2 The Taylor Expansion of Distributions.- 3.3 The Moment Asymptotic Expansion.- 3.4 Expansions in the Space P?.- 3.5 Laplace’s Asymptotic Formula.- 3.6 The Method of Steepest Descent.- 3.7 Expansion of Oscillatory Kernels.- 3.8 Time-Domain Asymptotics.- 3.9 The Expansion of f (?x) as ? ? ? in Other Cases.- 3.10 Asymptotic Separation of Variables.- 3.11 Exercises.- 4 Asymptotic Expansion of Multidimensional Generalized Functions.- 4.1 Introduction.- 4.2 Taylor Expansion in Several Variables.- 4.3 The Multidimensional Moment Asymptotic Expansion.- 4.4 Laplace’s Asymptotic Formula.- 4.5 Fourier Type Integrals.- 4.6 Time-Domain Asymptotics.- 4.7 Further Examples.- 4.8 Tensor Products and Partial Asymptotic Expansions.- 4.9 An Application in Quantum Mechanics.- 4.10 Expansion of Kernels of the Type f (?x, x).- 4.11 Exercises.- 5 AsymptoticExpansion of Certain Series Considered by Ramanujan.- 5.1 Introduction.- 5.2 Basic Formulas.- 5.3 Lambert Type Series.- 5.4 Distributionally Small Sequences.- 5.5 Multiple Series.- 5.6 Unrestricted Partitions.- 5.7 Exercises.- 6 Cesàro Behavior of Distributions.- 6.1 Introduction.- 6.2 Summability of Series and Integrals.- 6.3 The Behavior of Distributions in the (C) Sense.- 6.4 The Cesàro Summability of Evaluations.- 6.5 Parametric Behavior.- 6.6 Characterization of Tempered Distributions.- 6.7 The Space K?.- 6.8 Spherical Means.- 6.9 Existence of Regularizations.- 6.10 The Integral Test.- 6.11 Moment Functions.- 6.12 The Analytic Continuation of Zeta Functions.- 6.13 Fourier Series.- 6.14 Summability of Trigonometric Series.- 6.15 Distributional Point Values of Fourier Series.- 6.16 Spectral Asymptotics.- 6.17 Pointwise and Average Expansions.- 6.18 Global Expansions.- 6.19 Asymptotics of the Coincidence Limit.- 6.20 Exercises.- 7 Series of Dirac Delta Functions.- 7.1 Introduction.- 7.2 Basic Notions.- 7.3 Several Problems that Lead to Series of Deltas.- 7.4 Dual Taylor Series as Asymptotics of Solutions of Equations.- 7.5 Boundary Layers.- 7.6 Spectral Content Asymptotics.- 7.7 Exercises.- References.