Applied Asymptotic Analysis
About this Title
Peter D. Miller, University of Michigan, Ann Arbor, MI
Publication: Graduate Studies in Mathematics
Publication Year: 2006; Volume 75
ISBNs: 978-0-8218-4078-8 (print); 978-1-4704-1154-1 (online)
MathSciNet review: MR2238098
MSC: Primary 34-01; Secondary 34Exx, 35A35, 35Q53, 35Q55, 41A60
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.
Graduate students and research mathematicians interested in pure and applied mathematics and science and engineering.
Table of Contents
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Part 1. Fundamentals
Part 2. Asymptotic analysis of exponential integrals
- Chapter 2. Fundamental techniques for integrals
- 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
- Appendix: Fundamental inequalities