Elementary Real and Complex Analysis
By addebook • Jun 28th, 2008 • Category: Mathematics •
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Elementary Real and Complex Analysis (Dover Books on Mathematics)
Publisher: Dover Publications
Number Of Pages: 528
Publication Date: 1996-02-07
Sales Rank: 28817
ISBN / ASIN: 0486689220
EAN: 9780486689227
Binding: Paperback
Manufacturer: Dover Publications
Studio: Dover Publications
Average Rating: 5
Excellent undergraduate-level text offers coverage of real numbers, sets, metric spaces, limits, continuous functions, series, the derivative, higher derivatives, the integral and more. Each chapter contains a problem set (hints and answers at the end), while a wealth of examples and applications are found throughout the text. Over 340 theorems fully proved. 1973 edition.
Review:
It is one very interesting book
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.
Review:
An excellent pure maths text.
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.
Review:
Getting started in math analysis
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.
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