Applied Asymptotic Analysis

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.

Table of Contents

Part 1. Fundamentals

Chapter 1. The Nature of Asymptotic Appr

Chapter 2. Fundamental Techniques

Chapter 3. Laplace’s Method for Asymptotic Expansions of Integrals 

Chapter 4. The Method of Steepest Descents for Asymptotic  Expansions of Integrals 

Chapter 5. The Method of Stationary Phase for Asymptotic Analysis of Oscillatory Integrals 

Part 3. Asymptotic Analysis of Differential Equations

Chapter 6. Asymptotic Behavior of Solutions of Linear Second-order Differential Equations in the Complex Plane 

Chapter 7. Introduction to Asymptotics of Solutions of Ordinary Differential Equations with Respect to Parameters 

Chapter 8. Asymptotics of Linear Boundary-value Problems 

Chapter 9. Asymptotics of Oscillatory Phenomena 

Chapter 10. Weakly Nonlinear Waves 

Review

“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

About the Author

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