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Real and Complex Analysis (International Series in Pure and Applied Mathematics) [Hardcover]

Walter Rudin
4.6 out of 5 stars  See all reviews (25 customer reviews)

Price: $106.82 & FREE Shipping. Details
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Book Description

May 1, 1986 0070542341 978-0070542341 3
This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.


Frequently Bought Together

Real and Complex Analysis (International Series in Pure and Applied Mathematics) + Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) + Topology (2nd Edition)
Price for all three: $302.61

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Product Details

  • Hardcover: 483 pages
  • Publisher: McGraw-Hill Science/Engineering/Math; 3 edition (May 1, 1986)
  • Language: English
  • ISBN-10: 0070542341
  • ISBN-13: 978-0070542341
  • Product Dimensions: 6.6 x 0.9 x 9.6 inches
  • Shipping Weight: 1.7 pounds (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (25 customer reviews)
  • Amazon Best Sellers Rank: #151,572 in Books (See Top 100 in Books)

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Customer Reviews

It will enable one to get much more out of the book than slogging through it blindly. T. Sznigir  |  3 reviewers made a similar statement
Rudin's Real and Complex Analysis is an excellent book for several reasons. longhorn24  |  1 reviewer made a similar statement
I will inevitably be making a few comparisons between the two texts in the following. Mark Arjomandi  |  2 reviewers made a similar statement
Most Helpful Customer Reviews
105 of 106 people found the following review helpful
5.0 out of 5 stars A Comprehensive Guide to Analysis June 3, 2003
Format:Hardcover
Rudin's Real and Complex Analysis is an excellent book for several reasons. Most importantly, it manages to encompass a whole range of mathematics in one reasonably-sized volume. Furthermore, its problems are not mere extensions of the proofs given in the text or trivial applications of the results- many of the results are alternate proofs to major theorems or different theorems not proved. With that in mind, this book is not appropriate for a course where the instructor wants students to merely understand the theorems well enough to develop applications- the structure of the book is far better suited for a more theoretical course.

For example, the construction of Lebesgue measure is considered one of the most important topics in graduate analysis courses. After this construction, more abstract measures are developed, and then one proves the Riesz Representation Theorem for positive functionals later.

Conversely, Rudin develops a few basic topological tools, such as Urysohn's Theorem and a finite partition of unity, to construct the Radon measure needed in a sweeping proof of Riesz's Theorem. From this, results about regularity follow clearly, and the construction of Lebesgue measure involves little more than a routine check of its invariance properties.

Another example of where Rudin takes a more theoretical approach to provide a more elegant, yet less intuitive proof, is the Lebesgue-Radon-Nikodym theorem. Other books generally introduce signed measures with several examples, and use this result, along with properties of measures to derive the proof. On the other hand, since the first half of the book contains an intermission on Hilbert Space, Rudin uses the completeless of L^2 and the Riesz Representation Theorem for a more sweeping proof.

In the real analysis section, Rudin covers advanced topics generally not covered in a first course on measure theory. The chapters on differentiation and Fourier analysis are key examples of this. Rudin uses maximal functions to develop the Lebesgue Point theorem and results from complex analysis, and provides an incredibly thorough proof of the change-of-variables theorem. The ninth chapter, on Fourier transforms, relies heavily on convolutions, which are developed as a product of Fubini's theorem. This, in turn, is used to prove Plancherel's theorem and the uniqueness of Fourier transforms as a character homomorphism.

The tenth chapter, on basic complex analysis, essentially covers an entire undergraduate course on the subject, with added results based on a solid knowledge of topology on the plane. Once a solid foundation on the topic is laid, Rudin can develop more advanced topics from Harmonic analysis using general results from real analysis like the Hahn-Banach theorem and the Lebesgue Point theorem (for Poisson integrals).

Most of the basic results from the power series perspective are covered in the text, but while the geometric view is examined, it is still done in a very analytic, formula-based way that does not allow the reader to gain too much intuition. Nonetheless, all the basic results are covered, and Rudin uses these to develop the main theorems, such as the Mittag-Leffler and Weierstrass theorems on meromorphic functions, and the Monodromy Theorem and a modular function used to prove Picard's Little Theorem.

As an introductory text, even for advanced students, Rudin should probably be accompanied by more descriptive texts to develop better intuition. In fact, I would recommend Folland's Real Analysis and Ahlfors' Complex Analysis for self-study, because the problems are easier and one can learn better through those. With a good instructor, though, Rudin's text is concise and elegant enough to be both useful and enjoyable. It is also a good test to see how well one REALLY knows the subject.

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28 of 29 people found the following review helpful
By A Customer
Format:Hardcover
The first part of this book is a very solid treatment of introductory graduate-level real analysis, covering measure theory, Banach and Hilbert spaces, and Fourier transforms. The second half, equally strong but often more innovative, is a detailed study of single-variable complex analysis, starting with the most basic properties of analytic functions and culminating with chapters on Hp spaces and holomorphic Fourier transforms. What makes this book unique is Rudin's use of 20th-century real analysis in his exposition of "classical" complex analysis; for example, he uses the Hahn-Banach and Riesz Representation theorems in his proof of Runge's theorem on approximation by rational functions. At times, the relationship circles back; for example, he combines work on zeroes of holomorphic functions with measure theory to prove a generalization of the Weierstrass approximation theorem which gives a simple necessary and sufficient condition for a subset S of the natural numbers to have the property that the span of {t^n:n in S} is dense in the space of continuous functions on the interval. All in all, in addition to being a very good standard textbook, Real and Complex Analysis is at times a fascinating journey through the relationships between the branches of analysis.
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21 of 21 people found the following review helpful
5.0 out of 5 stars Persistence Pays. October 8, 2000
By Marete
Format:Hardcover
This Book of Rudin, Like Principles, rewards perhaps above all else, persistence; a virtue that, if we are to believe some professional Mathematicians, is indispensable for the study of Mathematics.

Its true that it is terse and efficient. However, this "short-coming" is to me not a short-coming at all for the simple reason that Rudin makes up for it. How? The problems. Once you get through the proofs, a TON of challenging questions will be waiting at the other end to hammer out of you any illusions about you depth of understanding. In my opinion, this is the greatest strenghth of Rudin's book. STICK with the problems, attack them relentlessly and at the end of it all, you will have learned, a little perhaps, how to think for yourself in Analysis.

As regards the section on Complex Variables, I found it fruitful to read it while supplementing the problems with those of Ahlfors, which is more computational (E.g. Although Rudin discusses complex int., he scarcely provides any problems for this, and the same goes for expansion in Power Series).

Stick with the book, and soon it will be like a classic novel. (At least it is for me)

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Most Recent Customer Reviews
5.0 out of 5 stars To paraphrase
"Do not stop reading [this] book. [Professor Rudin] does [things I became to appreciate]."

I hear Professor Rudin writes things down on pieces of paper and collects... Read more
Published 7 months ago by Miriosh
3.0 out of 5 stars Comprehensive and boring
If you need to study for a qualifying exam, get this book. If you want to see why analysis (especially complex analysis) is a beautiful subject, avoid like the plague. Read more
Published on December 2, 2010 by Narada
5.0 out of 5 stars Great Learning Experience
This is truly a well-crafted book. The organization is tight and the book is largely self-sufficient, really only calling upon material covered in his previous book, Principles of... Read more
Published on January 27, 2010 by T. Sznigir
5.0 out of 5 stars One of a Kind
I normally don't review books that already have this many reviews, especially when I agree so much with the reviews that already exist. Read more
Published on March 13, 2008 by Christopher Grant
4.0 out of 5 stars Rewarding, but if I had to choose one book...this would not be it
This is a very nice book. However in my opinion it is not the best of Rudin's three well-known books on analysis (Principles, Real and Complex, and Functional). Read more
Published on December 9, 2007 by Lucius Schoenbaum
5.0 out of 5 stars I love this book!
I love this book, even though I have not absorbed more than a small portion of it yet. I find this to be a much better book than the "baby Rudin", which struck me as dry, overly... Read more
Published on November 7, 2006 by Alexander C. Zorach
5.0 out of 5 stars My 2 cents
There are some excellent reviews here for this outstanding book, so I will try to avoid repetition. In preparation for my qualifying exams in graduate school, two of my colleagues... Read more
Published on October 11, 2006 by a reader
4.0 out of 5 stars Necessary, Necessary, Necessary
While I would not recommend this text to someone wishing to teach herself real and complex analysis, having this book in your personal mathematics library is a must for anyone... Read more
Published on July 23, 2006 by Geedess
5.0 out of 5 stars Real and Complex Analysis (Higher Mathematics Series)
The approach in this book is formal, yet not intuitive and neither natural for a beginning graduate student who have yet developed some level of mathematical maturity. Read more
Published on March 2, 2006 by Tru - a mathematician
5.0 out of 5 stars A start in math.
I am a fan of Rudin's books. This one "Real and Complex Analysis" has served as a standard textbook in the first graduate course in analysis at lots of universities in... Read more
Published on September 21, 2004 by Palle E T Jorgensen
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