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Applied Asymptotic Analysis

Peter D. Miller University of Michigan, Ann Arbor, MI
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Hardcover ISBN: 978-0-8218-4078-8
Product Code: GSM/75
List Price: $80.00 MAA Member Price:$72.00
AMS Member Price: $64.00 Electronic ISBN: 978-1-4704-1154-1 Product Code: GSM/75.E List Price:$75.00
MAA Member Price: $67.50 AMS Member Price:$60.00
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List Price: $120.00 MAA Member Price:$108.00
AMS Member Price: $96.00 Click above image for expanded view Applied Asymptotic Analysis Peter D. Miller University of Michigan, Ann Arbor, MI Available Formats:  Hardcover ISBN: 978-0-8218-4078-8 Product Code: GSM/75  List Price:$80.00 MAA Member Price: $72.00 AMS Member Price:$64.00
 Electronic ISBN: 978-1-4704-1154-1 Product Code: GSM/75.E
 List Price: $75.00 MAA Member Price:$67.50 AMS Member Price: $60.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$120.00 MAA Member Price: $108.00 AMS Member Price:$96.00
• Book Details

Volume: 752006; 467 pp
MSC: 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.

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 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

• 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 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 752006; 467 pp
MSC: 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.

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 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
• 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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.