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Real Analysis with Real Applications by Allan P. Donsig,Kenneth R. Davidson
  • Author: Allan P. Donsig,Kenneth R. Davidson
  • Title: Real Analysis with Real Applications
  • Size PDF ver: 1723 kb
  • Size ePUB ver: 1207 kb
  • Size Fb2 ver: 1686 kb
  • ISBN: 0130416479
  • ISBN13: 978-0130416476
  • Pages: 624
  • Other formats: lrf rtf docx azw
  • Category: Science & Math
  • Subcat: Mathematics
  • Language: English
  • Rating: 4.3 of 5
  • Votes: 518
  • Publisher: Prentice Hall (December 30, 2001)
  • Hardcover: Here
Real Analysis with Real Applications
Using a progressive but flexible format, this book contains a series of independent chapters that show how the principles and theory of real analysis can be applied in a variety of settings—in subjects ranging from Fourier series and polynomial approximation to discrete dynamical systems and nonlinear optimization. Users will be prepared for more intensive work in each topic through these applications and their accompanying exercises. Chapter topics under the abstract analysis heading include: the real numbers, series, the topology of R^n, functions, normed vector spaces, differentiation and integration, and limits of functions. Applications cover approximation by polynomials, discrete dynamical systems, differential equations, Fourier series and physics, Fourier series and approximation, wavelets, and convexity and optimization. For math enthusiasts with a prior knowledge of both calculus and linear algebra.

Reviews num: (7)

I'm now almost a third year in graduate school and searched Amazon for this book, as it was my favorite analysis textbook as an undergraduate, and I thought of buying it for reference to go with the solutions that I wrote (and saved) as an undergraduate.

I was shocked to find such poor reviews of such a well written text. I found the book very readable, the examples helpful, and most of all, the exercises very interesting and fun to solve.

Do not be put off by the previous critiques. This is an excellent book and it is the first book in analysis that enjoyed learning from.
Good book but concise in coverage of the basics, like the other reviewer said it gets to applications quickly(which is good) but requires mathematical maturity to use. Not a good book for someone who is taking their first proofs-class as it does not really hold your hand(though it is much more understandable than some of the classics of legendary difficulty, like Rudins PMA).

One downside is that it doesn't really cover multivariable analysis, there is a short section on vector valued functions in the application sections but that is all. Multivariable analysis isn't probably that important though, many books skip it.
I used this for an upper level undergraduate course in real analysis. The book is split about 50-50 on basic groundwork of analysis and then application (as the title would suggest). I found the text to be enjoyable to read for the most part. There are a few typos not addressed in the authors errata, particularly in the exercises, but for the most part very sleight. The presentation and proofs of results I found particularly nice. Often times the authors present proofs that are intuitive to visualize rather than proofs that make use of high-level machinery that too often obscures the intuition behind results (as some texts like Rudin do). This does at times lead to some begrudging pushing around of deltas and epsilons though.

The book does include some interesting examples to illustrate some important points, though at times I found my self wanting a few more examples just to put results into perspective. There are plenty of exercises, many of which are notably difficult.

One thing to note is that this book is probably not the best choice for a first course on analysis. There are many books that focus much more on the basics where as this one gets them out of the way pretty quickly and moves on the bigger and better things.
My college uses this text for the undergraduate real analysis sequence and I found it more helpful than the textbooks for most of my other math courses. Its examples are not as illustrative as Ross's and the proofs are not as concise as Rudin's, but I think Davidson and Donsig found a good balance between rigor and intuition. Unlike many of its competitors, this book provides a rigorous development of the theory of calculus without handwaivy omissions.
The first part of this book is a general introduction to the basics of real analysis, in which the authors try to strike a balance between rigor and intuition. What you get is a mixture of rigorous proofs with insufficient explanation and explanation with insufficient rigor to back it up. The text is often awkwardly worded and organized and notationally cumbersome for an intro text. If your looking for an intro text, there are much better options out there.

However, the second part of the book is a survey of various applications of real analysis and is very good. The scope of material is great ranging from dynamical systems to differential equations to wavelets to optimization. Each chapter is fascinating. There are 2 chapters on fourier series, one with applications to physics, one with approximations. I also find that the organization and overall style of the exposition works better in the second half.
I am currently a graduate student, and we are using this book in my first-year graduate course in analysis. To be quite honest, I find this book utterly useless, except for looking up homework problems that are so hard you are forced to look elsewhere just to learn how to solve them! The authors spend way too little time building up the theory and just expect their readers to be able to follow what they're doing with very few examples (or ones too complicated to really illustrate what's going on), and then give problems where even the easier ones can seem near impossible. This book makes more sense as a graduate text, certainly, especially if you've already had analysis; in that case, then you may only need to see the major theorems as a refresher and then you can start right on the challenging problems.

However, if you're an undergrad and this is your first exposure to analysis, go elsewhere, please! My fellow grad students and I have gotten so frustrated over this book and its problems, and we've all had analysis before! If you've never had analysis before, I would suggest Bartle/Sherbert's Intro. to Real Analysis; they spend a good amount of time with examples and what I call "warm-up" homework problems to get you used to the concepts, followed by some doozies (and, yes; selected answers and hints are in the back!). If you're very strong in math, then perhaps Rudin's "Principles of Mathematical Analysis" may be more up your alley (aka Baby Rudin). Best of luck to you!
A well balanced book! The first solid analysis course, with proofs, is central in the offerings of any math.-dept.;-- and yet, the new books that hit the market don't always hit the mark: The balance between theory and applications, --between technical profs and intuitive ideas,--between classical and modern subjects, and between real life exercises vs. the ones that drill a new concept. The Davidson-Donsig book is outstanding, and it does hit the mark. The writing is both systematic and engaged.- Refreshing! Novel: includes wavelets, approximation theory, discrete dynamics, differential equations,
Fourier analysis, and wave mechanics.

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