Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory (v. 1)

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Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory (v. 1) [Hardcover]

Carl M. Bender , Steven A. Orszag
4.7 out of 5 stars  See all reviews (22 customer reviews)

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Book Description

October 29, 1999 0387989315 978-0387989310 1999
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.

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Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory (v. 1) + Mathematical Methods for Physics and Engineering: A Comprehensive Guide
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Editorial Reviews


"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) 

Product Details

  • Hardcover: 593 pages
  • Publisher: Springer; 1999 edition (October 29, 1999)
  • Language: English
  • ISBN-10: 0387989315
  • ISBN-13: 978-0387989310
  • Product Dimensions: 1.3 x 6.3 x 9.4 inches
  • Shipping Weight: 2.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (22 customer reviews)
  • Amazon Best Sellers Rank: #517,690 in Books (See Top 100 in Books)

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Customer Reviews

4.7 out of 5 stars
4.7 out of 5 stars
Most Helpful Customer Reviews
32 of 33 people found the following review helpful
5.0 out of 5 stars The Classic Text on Asymptotics December 8, 2005
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.
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34 of 39 people found the following review helpful
5.0 out of 5 stars The Best Mathematical Lore Book Of All Time August 25, 2000
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!
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10 of 10 people found the following review helpful
5.0 out of 5 stars Remarkably powerful and practical tools for analysis November 30, 1996
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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7 of 7 people found the following review helpful
5.0 out of 5 stars Classic, fun to read April 19, 2011
Format:Hardcover|Verified Purchase
This is one of those books that you can read for fun. I'd rate it as my best book along with Carrier's ODE book. It says it's meant for advanced undergrads and grad students. Be that as it may, I could have read it in high school (mind you, its not trivial, but the material is so accessible to anyone with a basic knowledge of calculus that you can read it in high school). Also, the usefulness of it cannot be overstated. I owe my PhD to this book. Not having taken a formal asymptotics course, I had to read up perturbation theory to set up large activation asymptotics problems in combustion. The perturbation analysis was used in making sense of (sigh) hero simulations, and the degree of accuracy of these models was just remarkable.

Apart from the general high level of quality of examples and problems, the book almost literally talks to you. Beautiful phrases, such as 'executing several sleazy manoeuvres' and Holmes references are among those things that are perpetually endearing. Although I don't do any math anymore I still read this book a lot - on the BART train, or on a lazy Sunday afternoon. Very fun problems too. But best of all is that this book is genuinely useful to an idiot engineer. It exposes the power and beauty of perturbation theory in a way that you can look at a real problem and say 'Ah, it is simple'. I've spent many hours dreaming about things such as the interface between light and shadows, boundary layers (yes! the 'abrupt' transition from U_infty to 0 at the wall), why we need implicit formulations near the wall, and general considerations about stiff equations. This book is basically the vehicle that transports you there, and with minimum fuss.
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12 of 16 people found the following review helpful
4.0 out of 5 stars Very useful May 10, 2002
Very useful, in contrast with the cookbooks on 'mathematical physics' otherwise available. Systematic, with good examples and workable but challenging homework problems (I used this book for years while teaching advanced 'math methods in physics')Where else do you learn about Fuch's theorem? Good, systematic approach to asmptotic expansions, especially evaluation of integrals. Provided examples of deterministic chaotic systems before that subject was in vogue. Best description (as motion in phase space) of torque-free rigid body in free fall. Also good on boundary layer theory. Excellent text!
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2 of 2 people found the following review helpful
5.0 out of 5 stars Singularity is almost invariably a clue July 29, 2012
By ubpdqn
This is, perhaps, one of the best textbooks I have ever read. I have a habit of reading books that are far above my education and skills. There are multiple motivations to this approach: obtaining insight into areas of interest, learning from the introductory aspects and inspiration to learn more when the book and my understanding start to rapidly diverge.

This book is no exception with respect to this habit. However, the authors clarity and coherence exceeds many others I have read. In particular, the examples used are simple enough to be easily understood but powerful enough to illustrate the authors point, particularly when used as counterexamples or to show a limitation of a particular technique. The emphasis is on understanding and `seeing for yourself' what happens with various techniques and where they fall down. The authors also openly expose common misconceptions or pitfalls in thinking, particularly with respect to what is convergence, the importance of divergent series, the rendering meaningful of summation of divergent series.

Every chapter starts with motivating examples and builds to increasingly complex cases. Connections between areas in the book and limitations of analogies (e.g differential and difference equations) are clearly discussed. It was particularly interesting to see the connection between Pade approximants and continued fractions.

Finally, the quotations from Sherlock Holmes were delightful entres (and well chosen for chapter content).

I am left wishing I had the time to do this textbook justice and encouraged to start to fill the enormous gaps in my knowledge the book made clear to me.
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Most Recent Customer Reviews
4.0 out of 5 stars Thank you
Every thing in the book and delivery was perfect exept that it has no hard cover, I thought its price for hard cover.
Published 21 months ago by Dhuha Shareef
5.0 out of 5 stars A must own book but very challenging....
I was introduced to this book when I first took a course on Perturbation and Asymptotic Methods. The chapters on Boundary Layer theory, and WKB theory are well written and fun to... Read more
Published 22 months ago by Bessel
5.0 out of 5 stars Very good book
I used this book for a Graduate level course. I liked it a lot, is very good but a little bit dense. I still used it after the course, it is absolutely worth the price.
Published on March 16, 2011 by Manu
4.0 out of 5 stars useful book
The book is good and the methods seem effective. It assumes a very serious mathematical background, however, and the problems are quite challenging.
Published on February 19, 2011 by kgj
5.0 out of 5 stars Every home should have one
I have learned a lot of asymptotic methods from this book, having spent several previous years wondering what that stationary phase method that the physicsts were on about was (by... Read more
Published on April 21, 2010 by Narada
1.0 out of 5 stars Very disappointed.
I had seen this book in a bookstore over 15 years ago and regretted that I did not buy it at that time. Now that I have it, I am very disappointed. Read more
Published on February 9, 2010 by A. I. Haque
5.0 out of 5 stars outstanding!
I was looking for a long time for a textbook to use for a graduate course in advanced mathematics for engineers and this book is exactly what I need! Read more
Published on December 30, 2009 by CSD
5.0 out of 5 stars Still the Best
I used this book in grad school about 15 years ago and got it out again yesterday to learn more about sophisticated path deformations in the complex plane. Read more
Published on May 15, 2008 by T.I.
5.0 out of 5 stars Excellent !!!!!!!!
This is one of the finest books ever written on classical asymptotic analysis, one of the most useful areas in mathematics for engineering applications. Read more
Published on February 8, 2008 by Venkatram Krishnan
5.0 out of 5 stars very good
It's a very good book for students and engineers of science and technology. It's worth reading Orszag's book.
Published on October 25, 2007 by Weiguang Yang
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