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Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, by Carl M. Bender, Steven A. Orsza
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A clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. Aimed at teaching the most useful insights in approaching new problems, the text avoids special methods and tricks that only work for particular problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions.
- Sales Rank: #448063 in Books
- Published on: 1999-10-29
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x 1.44" w x 6.14" l, 2.11 pounds
- Binding: Hardcover
- 593 pages
Review
"This book is a reprint of the original published by McGraw-Hill \ref [MR0538168 (80d:00030)]. The only changes are the addition of the Roman numeral I to the title and the provision of a subtitle, "Asymptotic methods and perturbation theory". This latter improvement is much needed, as the original title suggested that this was a teaching book for undergraduate scientists and engineers. It is not, but is an excellent introduction to asymptotic and perturbation methods for master's degree students or beginning research students. Certain parts of it could be used for a course in asymptotics for final year undergraduates in applied mathematics or mathematical physics.
This is a book that has stood the test of time and I cannot but endorse the remarks of the original reviewer. It is written in a fresh and lively style and the many graphs and tables, comparing the results of exact and approximate methods, were in advance of its time. I have owned a copy of the original for over twenty years, using it on a regular basis, and, after the original had gone out of print, lending it to my research students. Springer-Verlag has done a great service to users of, and researchers in, asymptotics and perturbation theory by reprinting this classic." (A.D. Wood, Mathematical Reviews)
Most helpful customer reviews
41 of 42 people found the following review helpful.
The Classic Text on Asymptotics
By Glenn Ledder
I am a mathematics professor, and asymptotic methods is my area of expertise. Bender and Orszag was the standard text on this topic in the early 1980s, and it remains by far the most thorough book on the subject. I had two courses from it in graduate school and have taught from it 3 times now. I have never found a mistake. The book is very well written; its only weakness is that its graphics reflect the technology of the 1970s. Nevertheless, I am teaching from it again next year, because it is in a class by itself.
There is one thing that individual readers and faculty users should be aware of. Some of the exercises, including a few marked as "intermediate," are incredibly difficult. My instructor made the mistake of assigning exercises without working them first. I am careful not to assign any exercises until after I have worked them.
37 of 42 people found the following review helpful.
The Best Mathematical Lore Book Of All Time
By Philip Hobbs
I learned more mathematics from Bender & Orszag than from any other math book I own. I'm an applied physicist, and as any physicist knows, a sleazy approximation that provides good physical insight into what's going on in some system is far more useful than an unintelligible exact result.
This book covers approximate methods for solving differential and difference equations, asymptotic methods for integrals, and asymptotic and extrapolation methods for sums. There are a great many beautiful plots, and lots of discussion of the actual lore of doing--alternative ways of attacking the same problem, things to watch out for, what sorts of problems a given method is best at.
I think the most valuable parts of this book are the examples and problems, both of which are the best anywhere.
It's really great to see this old friend (first published in 1978) back in print. If you have ugly differential equations or integrals to solve, buy it!
11 of 11 people found the following review helpful.
Remarkably powerful and practical tools for analysis
By A Customer
This is the best applied mathematics book I have ever read, bar none. It is primarily a text on asymptotic methods for integrals, sums, differential equations, and difference equations (recurrence relations). Major topics are Laplace's method, steepest descents, stationary phase, WKB theory, boundary layer theory, and regular and singular perturbation theory. The exposition shows an understanding of the needs and tastes of physicists and engineers far beyond the usual dry theorems. It is full of lore and insight, plus the best analytic and numerical examples extant. There are hundreds of good problems as well. If you need to use asymptotic methods in your work, or just want a good read in applied math, this is your book
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