Asymptotic Methods in Analysis (Dover Books on Mathematics) by N. G. de Bruijn (PDF)

Description

A reader looking for interesting problems tackled often by highly original methods, for precise results fully proved, and for procedures fully motivated, will be delighted. — Mathematical Reviews.Asymptotics is not new. Its importance in many areas of pure and applied mathematics has been recognized since the days of Laplace. Asymptotic estimates of series, integrals, and other expressions are commonly needed in physics, engineering, and other fields. Unfortunately, for many years there was a dearth of literature dealing with this difficult but important topic. Then, in 1958, Professor N. G. de Bruijn published this pioneering study. Widely considered the first text on the subject — and the first comprehensive coverage of this broad field — the book embodied an original and highly effective approach to teaching asymptotics. Rather than trying to formulate a general theory (which, in the author’s words, “leads to stating more and more about less and less”) de Bruijn teaches asymptotic methods through a rigorous process of explaining worked examples in detail.Most of the important asymptotic methods are covered here with unusual effectiveness and clarity: “Every step in the mathematical process is explained, its purpose and necessity made clear, with the result that the reader not only has no difficulty in following the rigorous proofs, but even turns to them with eager expectation.” (Nuclear Physics).Part of the attraction of this book is its pleasant, straightforward style of exposition, leavened with a touch of humor and occasionally even using the dramatic form of dialogue. The book begins with a general introduction (fundamental to the whole book) on O and o notation and asymptotic series in general. Subsequent chapters cover estimation of implicit functions and the roots of equations; various methods of estimating sums; extensive treatment of the saddle-point method with full details and intricate worked examples; a brief introduction to Tauberian theorems; a detailed chapter on iteration; and a short chapter on asymptotic behavior of solutions of differential equations. Most chapters progress from simple examples to difficult problems; and in some cases, two or more different treatments of the same problem are given to enable the reader to compare different methods. Several proofs of the Stirling theorem are included, for example, and the problem of the iterated sine is treated twice in Chapter 8. Exercises are given at the end of each chapter.Since its first publication, Asymptotic Methods in Analysis has received widespread acclaim for its rigorous and original approach to teaching a difficult subject. This Dover edition, with corrections by the author, offers students, mathematicians, engineers, and physicists not only an inexpensive, comprehensive guide to asymptotic methods but also an unusually lucid and useful account of a significant mathematical discipline.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Although I think the material is reasonably explained, I do think there is a disconnect from the exercises against the material’s level of discussion.

⭐It has comprehensive coverage…but I still think the author is too abstract. Unfortunately, its one of the only books out there….but search the internet…you can find really good University class notes now. That will supplement this book.

⭐The book is like a David Attenborough animal show: at every turn there is a new marvelous “animal” that pops its head out. What makes this possible is the subject: building approximations using asymptotic methods. Each remarkable approximation comes after the author has showed us how to almost painlessly ferret it out. This is why the book works so well, chapter after chapter we learn beautiful tricks through a clear and concise presentation.

⭐a classic…

⭐this book stands apart on my shelves since I used it so often that I had to replace it three times; the subject of asymptotics is both useful and fascinating; this book delves in it through many examples first showing how to deal with them in elementary fashion, then using more advanced tools and finally exhibiting heavy machinery; although a lot of complex analysis tools are present in the text, it is quite possible to get a good grasp of what is done in it since elementary methods are always stated in the first instance e.g. the study of U_n+1 = sin(U_n): using calculus, it is shown that U_n which tends to zero is not far from (3/n)^(1/2) (sorry about that for those who do not speak fluent mathematics…), then 4 terms of the asymptotic development are given followed by a more general method to get an arbitrary number or terms.This book will prove useful to you in different fashion as the years go onand your mathematical skill is improving.Moreover, the price is really affordable.As a matter of fact, I first bought this book in 1986…and I find it still really pleasurable.

⭐This is a Classic introduction to Asymptotics. Its main strength, is the illustration of techniques through the detailed working out of examples. In addition, there is FULL rigour, so that the proofs are complete.To study this book you should be comfortable with:a) Undergraduate Level Real Analysis. b) Elementary Notions from Complex Variable Theory including Complex Integration, Calculus of Residues, Power Series Expansions and related ideas.

⭐This will be an alternative review since I lack the complete knowledge foundation required by the book, yet I’ve purchased it knowing so in advance.This is simply an elegantly written book in terms of language use. The book is designed with a language use; the way i prefer to write swedish text, no matter the subject at hand.Also parsing arbitrarly there are some references to other russians work. As far as I can evaluate by glimpse and rudimentary mathematical knowledge there is proof and definition driven reasoning which is a must in mathematics in order to be of great use for the reader at hand.Also for a novice like myself the extensive and clear, instant writing about necessary mathematical methods, functions and so on to solve the problems developed, functions in use give back great insight value; The art of thinking of how go about solving the issues at hand.I think one would feel confident in reading this as one go by reading mathematical/Physics courses at University (if you do/did). This mean that for me the reading would be limited to pieces of reading as one grow with required knowledge.10-20 points in mathematical analysis and additional introduction course in functions of complex variables would suffice (of course this is an estimation).

⭐I think this book will provide finishing touches to the aspiring mathematical physicist.

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