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Hardcover ISBN:  9780821840788 
Product Code:  GSM/75 
List Price:  $135.00 
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AMS Member Price:  $108.00 
Sale Price:  $87.75 
eBook ISBN:  9781470411541 
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eBook ISBN:  9781470411541 
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Book DetailsGraduate Studies in MathematicsVolume: 75; 2006; 467 ppMSC: Primary 34; 41; 74; 33; 81;
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.
ReadershipGraduate students and research mathematicians interested in pure and applied mathematics and science and engineering.

Table of Contents

Part 1. Fundamentals

Chapter 0. Themes of asymptotic analysis

Chapter 1. The nature of asymptotic approximations

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 secondorder 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 boundaryvalue problems

Chapter 9. Asymptotics of oscillatory phenomena

Chapter 10. Weakly nonlinear waves

Appendix: Fundamental inequalities


Additional Material

Reviews

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 studentfriendly 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 studentfriendly ... 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 wellwritten, 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


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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.

Part 1. Fundamentals

Chapter 0. Themes of asymptotic analysis

Chapter 1. The nature of asymptotic approximations

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 secondorder 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 boundaryvalue problems

Chapter 9. Asymptotics of oscillatory phenomena

Chapter 10. Weakly nonlinear waves

Appendix: Fundamental inequalities

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 studentfriendly 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 studentfriendly ... 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 wellwritten, 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