**Graduate Studies in Mathematics**

Volume: 75;
2006;
467 pp;
Hardcover

MSC: Primary 34; 41; 74; 33; 81;

Print ISBN: 978-0-8218-4078-8

Product Code: GSM/75

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-1154-1
Product Code: GSM/75.E**

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

#### Supplemental Materials

# Applied Asymptotic Analysis

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*Peter D. Miller*

This book is a survey of asymptotic methods set in the current applied
research context of wave propagation. It stresses rigorous analysis in
addition to formal manipulations. Asymptotic expansions developed in the
text are justified rigorously, and students are shown how to obtain solid
error estimates for asymptotic formulae. The book relates examples and
exercises to subjects of current research interest, such as the problem of
locating the zeros of Taylor polynomials of entire nonvanishing functions
and the problem of counting integer lattice points in subsets of the plane
with various geometrical properties of the boundary.

The book is intended for a beginning graduate course on asymptotic
analysis in applied mathematics and is aimed at students of pure and
applied mathematics as well as science and engineering. The basic
prerequisite is a background in differential equations, linear algebra,
advanced calculus, and complex variables at the level of introductory
undergraduate courses on these subjects.

The book is ideally suited to the needs of a graduate student who, on the one hand, wants to learn basic applied mathematics, and on the other, wants to understand what is needed to make the various arguments rigorous. Down here in the Village, this is known as the Courant point of view!!

—Percy Deift, Courant Institute, New York

Peter D. Miller is an associate professor of mathematics at the University
of Michigan at Ann Arbor. He earned a Ph.D. in Applied Mathematics from the
University of Arizona and has held positions at the Australian National
University (Canberra) and Monash University (Melbourne). His current research
interests lie in singular limits for integrable systems.

#### Readership

Graduate students and research mathematicians interested in pure and applied mathematics and science and engineering.

#### Reviews & Endorsements

The new book by Peter Miller is a very welcome addition to the literature. As is to be expected from a textbook on applied asymptotic analysis, it presents the usual techniques for the asymptotic evaluation of integrals and differential equations. It does so in a very clear and student-friendly way. The methods are first introduced at an informal level, which enables students to understand the main ideas. It also serves as motivation for more technical details. Rigorous proofs and estimates can be very tedious but they are carefully presented. ... What is really special about the book is that it includes discussions on a number of topics that are usually not found in books on asymptotics, since it is assumed that students have seen it in other courses. ... The inclusion of these topics at the places where they are needed is an extra bonus which greatly adds to the usefulness of the book. ... Peter Miller's book is an ideal textbook for a graduate course on asymptotic analysis. Highly recommended.

-- Journal of Approximation Theory

What is really special about the book is that it includes discussions on a number of topics that are usually not found in books on asymptotics ... very clear and student-friendly ... ideal textbook for a graduate course on asymptotic analysis. Highly recommended.

-- Arno Kuijlaars for Journal of Approximation Theory

This manuscript will definitely have a big impact in showing that applied asymptotics analysis derives from classical analysis. Moreover, applications continue to demonstrate its continuing importance and vitality. Miller does an outstanding job of delivering this important message.

-- Robert O'Malley, University of Washington

This book is very well-written, is mathematically very careful, and he has done a terrific job in explaining many of the subtle points in asymptotic analysis ... the quality is certainly first rate. ... His pedagogy is excellent.

-- Michael Ward, University of British Columbia

This book combines some of the best information available to graduate students on asymptotics...Miller seems to add lots of motivation and careful explanations, certainly indicating that he was a top students and that he is a good teacher. ... In summary, this new book brings one to the frontier of much current research, both pure and applied. ... I recommend it highly.

-- SIAM Review

Peter Miller's book is an ideal textbook for a graduate course on asymptotic analysis. Highly recommended.

-- Journal of Approximation Theory

#### Table of Contents

# Table of Contents

## Applied Asymptotic Analysis

Table of Contents pages: 1 2

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface xiii14 free
- Part 1. Fundamentals 118 free
- Chapter 0. Themes of Asymptotic Analysis 320
- Chapter 1. The Nature of Asymptotic Approximations 1532

- Part 2. Asymptotic Analysis of Exponential Integrals 4562
- Chapter 2. Fundamental Techniques for Integrals 4764
- Chapter 3. Laplace's Method for Asymptotic Expansions of Integrals 6178
- §3.1. Introduction 6178
- §3.2. Nonlocal Contributions 6279
- §3.3. Contributions from Endpoints 6481
- §3.4. Contributions from Interior Maxima 6784
- §3.5. Summary of Generic Leading-order Behavior 7087
- §3.6. Application: Weakly Diffusive Regularization of Shock Waves 7390
- §3.7. Multidimensional Integrals 87104
- §3.8. Notes and References 93110

- Chapter 4. The Method of Steepest Descents for Asymptotic Expansions of Integrals 95112
- §4.1. Introduction 95112
- §4.2. Contour Deformation 97114
- §4.3. Paths of Steepest Descent 98115
- §4.4. Saddle Points 103120
- §4.5. Parametrization-independent Local Contributions 107124
- §4.6. Application: Long-time Asymptotic Behavior of Diffusion Processes 108125
- §4.7. Application: Asymptotic Behavior of Special Functions, Airy Functions and the Stokes Phenomenon 116133
- 4.7.1. Integral representations for Airy functions 116133
- 4.7.2. Preliminary transformations necessary for asymptotic analysis of Ai(x) for large x 117134
- 4.7.3. Determination of the path. Dependence of the path on k 119136
- 4.7.4. Asymptotic behavior of Ai(x) for large x. The Stokes phenomenon 122139

- §4.8. The Effect of Branch Points 125142
- §4.9. Notes and References 147164

- Chapter 5. The Method of Stationary Phase for Asymptotic Analysis of Oscillatory Integrals 149166
- §5.1. Introduction 149166
- §5.2. Nonlocal Contributions 151168
- §5.3. Contributions from Interior Stationary Phase Points 156173
- §5.4. Summary of Generic Leading-order Behavior 162179
- §5.5. Application: Long-time Behavior of Linear Dispersive Waves 164181
- §5.6. Application: Semiclassical Dynamics of Free Particles in Quantum Mechanics 171188
- 5.6.1. Derivation of the dispersion relation for "matter waves" 171188
- 5.6.2. The Schrodinger equation for a free particle. Interpretation of the Schrodinger wave function 173190
- 5.6.3. The semiclassical limit. Heuristic reasoning 174191
- 5.6.4. Rigorous semiclassical asymptotics using the method of stationary phase 177194

- §5.7. Multidimensional Integrals 181198
- §5.8. Notes and References 193210

- Part 3. Asymptotic Analysis of Differential Equations 195212
- Chapter 6. Asymptotic Behavior of Solutions of Linear Second-order Differential Equations in the Complex Plane 197214
- Chapter 7. Introduction to Asymptotics of Solutions of Ordinary Differential Equations with Respect to Parameters 253270
- §7.1. Regular Perturbation Problems 254271
- §7.2. Singular Asymptotics 263280
- 7.2.1. The WKB method 263280
- 7.2.2. The special case of an asymptotic power series for f(x; λ) 268285
- 7.2.3. Turning points 277294
- 7.2.4. Problems with more than one turning point. The Bohr-Sommerfeld quantization rule 300317
- 7.2.5. Uniform asymptotics near turning points. Langer transformations 304321

- §7.3. Notes and References 310327

- Chapter 8. Asymptotics of Linear Boundary-value Problems 311328
- §8.1. Asymptotic Existence of Solutions 312329
- §8.2. An Exactly Solvable Boundary-value Problem: Phenomenology of Boundary Layers 315332
- §8.3. Outer Asymptotics 318335
- §8.4. Rescaling and Inner Asymptotics for Boundary Layers and Internal Layers 321338
- §8.5. Matching of Asymptotic Expansions, Intermediate Variables, and Uniformly Valid Asymptotics 325342
- §8.6. Examples 328345
- §8.7. Proving the Validity of Uniform Approximations 342359
- §8.8. The Method of Multiple Scales 350367
- §8.9. Notes and References 353370

- Chapter 9. Asymptotics of Oscillatory Phenomena 355372
- §9.1. Perturbation Theory in Linear Algebra and Eigenvalue Problems 356373
- §9.2. Periodic Boundary Conditions and Mathieu's Equation 368385
- §9.3. Weakly Nonlinear Oscillations 382399
- 9.3.1. Periodic solutions near equilibrium 383400
- 9.3.2. A perturbative approach to weak cubic nonlinearity. Secular terms 384401
- 9.3.3. Removal of secular terms. Strained coordinates and the Poincare-Lindstedt method 388405
- 9.3.4. The method of multiple scales 391408
- 9.3.5. Justification of the expansions 397414

- §9.4. Notes and References 400417

- Chapter 10. Weakly Nonlinear Waves 401418
- §10.1. Derivation of Universal Partial Differential Equations Using the Method of Multiple Scales 401418
- 10.1.1. Modulated wavetrains with dispersion and nonlinear effects. The cubic nonlinear Schrödinger equation 402419
- 10.1.2. Spontaneous excitation of a mean flow 410427
- 10.1.3. Multiple wave resonances 417434
- 10.1.4. Long wave asymptotics. The Boussinesq equation and the Korteweg-de Vries equation 423440
- §10.2. Waves in Molecular Chains 425442
- §10.3. Water Waves 433450
- §10.4. Notes and References 447464

- Appendix: Fundamental Inequalities 451468
- Bibliography 453470
- Index of Names 455472
- Subject Index 457474

Table of Contents pages: 1 2