A Distributional Approach to Asymptotics
Springer-Verlag New York Inc.
978-1-4612-6410-1 (ISBN)
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
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.
| Reihe/Serie | Birkhäuser Advanced Texts Basler Lehrbücher |
|---|---|
| Zusatzinfo | XV, 454 p. |
| Verlagsort | New York |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Analysis | |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| ISBN-10 | 1-4612-6410-3 / 1461264103 |
| ISBN-13 | 978-1-4612-6410-1 / 9781461264101 |
| Zustand | Neuware |
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