Customer Reviews

Elementary Real and Complex Analysis (Dover Books on Mathematics)

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This book by Shilov covers the fundamentals in beginning analysis(both real and complex). It has in common with Walter Rudin's book (entitled 'Real and Complex Analysis') that it covers both real functions (integration theory and more), as well as Cauchy's theorems for analytic functions. Shilov's book is at an undergraduate level, and it can easily be used for self-study. The Dover edition is affordable. Rudin's book is for the beginning graduate level, and it is widely used in math departments around the world. Both books have stood the test of time.
Comparison of Shilov with Rudin: Rudin's 'Real and Complex' has become an institution, and I have to admit I have loved it since I was a student myself, but conventional wisdom will have it that Shilov is a lot gentler on students, and much easier to get started with: It stresses motivation a bit more, the exercises are easier (some of Rudin's exercises are notorious, but I find the challenge charming--not all of my students do though!), and finally Shilov gets to touch upon a few applications; fashionable these days. But that part easily gets dated. I will expect that beginning students will enjoy Shilov's book.
Personally, I find that with perseverance, students who keep at it with Rudin's book, will end up with a lot stronger foundation. They are more likely to have proofs in their blood. I guess Shilov can always serve as a leisurely supplementary reading to Rudin.
There will never be another book like Rudin's 'Real and Complex', just like there will never be another van Gogh. But the fact that we love van Gogh doesn't prevent us from enjoying other paintings.

I purchased this book as a reference book for my first analysis course. It is very well written, and easy to follow. Dr. Shilov has a very nice way of organizing this text: He puts all the definitions at the beginning of the chapter and the subsequent sections are results of those definitions. It makes for a very quick reference. His presentation of the included proofs is also very nice. There were several occasions I found myself thumbing through it for a second perspecitve.

As far as the actual material presented, Dr. Shilov starts off with funtions of one real variable, then rather quickly generalizes to complex variables and N dimensional functions, so you'll quickly see metric theory and some topology. He does keep in mind this is intended for undergrads and first year grads though.

Oh, another nice feature is the price! I'd recommend this book to any math enthusiast as a reference, or to someone going through an early analysis course.

The book is one of the finest mathematics texts that I have ever read. That does'nt mean much because I'm 16 years old, and have only been studying advanced math for a little over a year. The treatment was basic enough for someone like me to understand. I strongly recommend it to college students and young people who are interested in mathematics.

After introducing concepts from set theory, the foundations of analysis, and the notion of a "mathematical structure," he gives a detailed presentation of limits and series. He also introduces elementary functions in terms of their functional equations. Then he covers differentiation and integration, first of real, and then of complex functions. He uses Taylor series to introduce ideas about complex functions.

In short, it is a good book for those who hope to become mathematicians, physicists, or engineers, and have had a few college math courses already.

Coherent and comprehensive, this "Elementary Real and Complex Analysis" is an emphatic introductory text, which will provide undergraduates with all the guides that they may need.
The presentation of this book is such that anybody who is taking Pure and/or Applied Mathematics course would value it. From Analysis Basics to Complex Functions, the authors of this book fulfilled every desire.
Worthy of mention is the way they simplified the rather complex Cauchy's Theorem. The same could be said of the chapters covering: Power Series Representations, Topology and Analysis in the Complex Plain, Holomorphic Functions, and Contour Integration.
Each topic that appeared in this book received accurate simplification. They are all object-oriented, and were designed to be of great use to self-learners.
Highly recommended!

I purchased this book to study some complex analysis. Being a physicist I would like to brush up on this. The book was completely different to what I expected.

Some applications would have been nice, but this text is pure maths. The book is well written, easy to follow and concise. I ended up reading it and gained and appreciation for the thorough consideration of elementary real and complex numbers.

Shilov is thorough and avoids making leaps and assertions. This would make the book readable to lower undergraduates. However the significance of some things is not explained, or explained in a very dry manner so people might miss this.

I highly recommend this book if you are interested in real and complex analysis from a pure mathematics perspective.

To me, the best chapters of this book are that about series and integrals. The text is plenty of interesting notions, like that of direction that is related with the notion of limit. I appreciated very much the study that Shilov does about parameter-dependent proper and improper integrals. The topological notions are placed in one intuitive manner. Without doubt, this is one very good and clear exposition about the subject. However, I think that the problems are not easy. Also sometimes Shilov states the theorems with additional conditions that are not useful. For example, this happens usually in the chapter about derivatives because the definition of derivative given by Shilov imposes that any function with derivative in the interval of the domain has continuous derivative in the interior points of its domain. However, Shilov charges some theorems with the extra condition of continuous derivative.
When the Taylor's formula is presented in page 252 - Theorem 8.22, it is stated that the error of the approximation is computed in some interior point of the interval, what is not completely correct. For example, take the second degree Taylor's approximation around x = 0 of the function x raised to the third power, and you will see that in this case the error is computed on one extreme point of the interval.
Also the proof of the theorem 10.49b (page 415) has logical problems of the kind that may arise during the translation.
However, these remarks are small questions without consequences for the course of the exposition.

Can't beat the price, and the material is well-presented and organized, but it's stripped down to the bare essentials - theorem, proof, lemma, corollary, etc. It's not a book on proof methodology, for sure. I graduated with a degree in computer science, but I haven't done a proof for a while and never took a class beyond linear algebra, and I wanted to teach myself analysis. While I don't find the material too difficult to follow, I really don't find it all that great for self-study. The book yields conclusion after conclusion, but among all the results, I find the text doesn't do a great job of conveying its methodology. In other words, the book spends the vast majority of the time developing new results (the "what" of analysis), but it does little to prepare the reader to understand the "how" of analysis. I feel as though the book is giving me a fish, rather than teaching me to fish.

And there are some idiosyncracies. You need to be wary of an occasionally swapped subscript, for instance. And in chapter 1 problem 5: Which is larger, Sqrt(3) + Sqrt(5) or Sqrt(2) + Sqrt(6)? The answer in the back of the book is plain wrong. And the book proves something as fundamental as the uniqueness of 1; and yet it invokes the binomial theorem out of the blue?

Anyway, the price is right, but beware that it might make a better reference or a collection of examples than a primary self-study guide. It's not that it's "too easy" as one reader put it; rather, it doesn't integrate the material with exercises and explanations well enough for my liking.

The dover books are inexpensive so they are a low risk investment. I have found they are hit and miss.

This one is a big hit.

As others have noted his presentation is straightforward. I wish it were a bit more instructive, like more examples and exercises, but that does not diminish the value.

The author packs alot into this book. His coverage of fundametnals in the beginning of the book is excellent. By chapter 3 he presents metric spaces. Which is where I am at so far.

It will probably take me a year to get through this book, but it will be a very rewarding year. From intervals to Cauchy, it will be a very rewarding year.

For my first foray into analysis, a math course I foolishly avoided in college, I picked this book because of all the good comments (and its low price). I want it to be good. But thoroughly working through an analysis book is an enormous investment of time and energy, so I want to invest in something solid. As of p. 12, I am already beginning to wonder whether this book is shot full of typos, which will drive me nuts. I am painfully aware of my own many stupid mistakes in math, so when I run into something weird, my first reaction is to assume I have made yet another mistake, not that the book is wrong. What scares me is that I may not be wrong here.

The first section is a somewhat tedious crawl through basics--axioms of addition, multiplication, ordering, etc. and their implications. I'm motivated to build from the ground up, so when I get to the famous Hanky-Panky theorem or whatever, I can think back and realize that the key to it all is that |xy| = |x||y|, or that x<=a if x<=0. So I crawl.

On page 12, I hit the following:
"THEOREM: The formula |xy| = |x||y| holds for all x and y.
Proof: ...Similarly, if x<=0, y<=0, then...xy<=0, by Theorem 1.55g..."
Huh?
[BTW, I hate his use of commas. Since when is a comma a symbol for a logical operator like "or" or "and"? Why not "if x<=0 and y<=0" if that's what he means? Was the comma due to Soviet Russian glorious proletarian restrictions on printing ink?]
The relevant part of Theorem 1.55g is "...if x<=0, y<=0, then xy>=0", which matches what I learned in junior high.

Shilov is constantly flipping back and forth between x>=y and y<=x, for no apparent good reason, which is irritating even when the statement is correct. But I wonder whether he cooked his own goose by missing a flip that he should have done.

I was already suspicious because on p. 10, he has a proof for his theorem 1.54.a:
"Proof. Obviously x<=a if x<=o... Moreover...-x = |x| <=a if x<=0."
Huh? Two completely opposite conclusions, x<=a and -x<=a, under the same condition, x<=0?

Do I need new glasses, or a new brain? Or did I get it right and did MIT Press, the original publisher, and Dover not notice or care about this kind of stuff? Are there typos till p. 12, then no more till the end? Is it worth sticking with this, or am I going to find typos in the proof of the Hanky-Panky theorem too? Should I invest in Rudin instead?

As Shilov write in the introduction "I have tried to accomodate the interests of larger population of those concerned with mathematics" and at that he seems to do. However, the book does require some mathematical background as he appears to omit defining a few things. I believe the book would be ideal for those who want a handy reference, or an easier book when struggling with an analysis course.

However, for the more mathematically inclined readers, the problems are often too easy, and many things are proved that could be better left as exercises. For a more difficult Analysis book, I would reccomend Rudin.