Book Description
This is the first text in a generation to re-examine the purpose of the mathematical statistics course. The book's approach interweaves traditional topics with data analysis and reflects the use of the computer with close ties to the practice of statistics. The author stresses analysis of data, examines real problems with real data, and motivates the theory. The book's descriptive statistics, graphical displays, and realistic applications stand in strong contrast to traditional texts that are set in abstract settings. ... Read more

Customer Reviews (31)

excellent text
This book got very mixed reviews from 1 star to 5. I am in agreement with Froese's review and give it 4 stars. Rice is trying to write a book for statistics students who are not mathematics or statistics majors without shortchanging them on the advanced topics and the theory. This can be difficult and often alienates both the beginners and those interested in advanced methods. I have tried to stay along that fine line with my texts also. So I appreciate the difficulties. As an author of a book on bootstrap methods, I also appreciate the way Rice has integrated that subject into this text.

Two and a half stars would probably be more accurate
My professor used this as a text for a graduate overview of mathematical statistics course, with no prior experience in statistics required.

However, this book is not for the neophyte in statistics, and would have DEFINITELY benefited from a accompanying manual with extra problems and FULLY WORKED SOLUTIONS. Needless to say the problems in the book, although plentiful, although informative, were not what I would consider easily approachable to non-statistical gurus.

Also, I felt the organization of the book could have been better. A section in each chapter clearly outlining the concepts would have been nice.

But, if you have a great professor, and other references on hand, then yes, you could make good use of this book.

Highly Recommended Book and Author
This is an absolutely fantastic book, written by one of the best professors I've ever had for a technical class. I took Prof. Rice's course, Stat 135, at UC Berkeley and it was a really great and illuminating experience. The book, the lectures, and the course overall, cemented my decision to become a Statistics major. The book is technical, but very accessible, and contains applications ranging from Astrophysics to Jane Austen to Finance to Biology and more. It is ideal for self study and exceedingly interesting to just sit down and read. The difficulty level of the problems is fairly distributed such that slow learners have a chance to acclimate themselves to the material, and the quicker ones of us can look to the later problems and be sufficiently challenged.

Don't believe the reviews here or the ones on [...]. They're just outright wrong. This is a great book written by a great professor. If it's over your head, then maybe you don't realize it is written for an upper division Mathematical Statistics course.

Outstanding: undoubtedly the best text written at this level
This book is an introductory-level statistics textbook for people who are learning statistics for the first time, and who know some math but are not math wizards.

I am outright shocked at the low reviews given to this book.I tend to be highly critical of statistics textbooks, especially ones at this level.In my opinion, this book does an outstanding job.It balances making the material accessible with going into considerable depth, and it integrates mathematical theory with an emphasis on data analysis.The title of this book says it all: "Mathematical Statistics and Data Analysis".There are few other books that emphasize both theory and practice.This book certainly lives up to its title in this respect.

Perhaps some of the bad reviews are from frustrated students who are in classes that are moving too quickly for their level.Just because this is an excellent book doesn't mean that every professor will use it in an appropriate way.The exercises in this book can be tough and I think it would be possible to easily overwhelm students by assigning too many problems or problems that are too difficult.This book is only really useful if you are able to explore it at a slower pace, reading it and reflecting on it.

---

This book is not concise but it is not overly wordy.Rice is an excellent writer.The expanded discussion communicates aspects of the subject that are often overlooked, and helps build the students' intuition.This book is always readable and it is clearly written to be read and understood.In addition to communicating basic concepts, the book also explores numerous practical and philosophical considerations.For a book at such an introductory level, this text is remarkably deep.

One of the most attractive aspects of this book is that it is relatively easy to skip around in it.More advanced students or students with some exposure to probability will necessarily want to skip many of the early chapters.Unlike some of the more advanced texts (such as the Casella and Berger) where the chapters depend on and reference each other in a rather rigid fashion, this book is more flexible and thus can appeal to students with diverse backgrounds.

My last bit of praise is the integration of data analysis with mathematical theory.Distributions are introduced with a rich discussion of where, how, and why they arise in practice.Data analysis techniques, tests, and abstract definitions alike are introduced in such a way that they are rooted both to the abstract theory and to consideration of applications: this book is one of the rare texts that fully bridges the gap between the real world and the abstract models used to describe it.

I would recommend this book for a wide variety of different uses.This would make an outstanding textbook and it could easily be used for more than one course, or for courses at more than one level.It is a useful book to have on the shelf as a basic reference, and it is also very useful for self-study.My only complaint is that more books in the field of statistics are not written in the spirit and style of this one.In particular, I would like to see authors of more advanced books examine this book's strengths and use them to enhance their own writing.Or perhaps John Rice ought to try his hand at writing a higher-level text; I know I would be eager to see what he could produce!

Decision Theory Gone?
From 2nd edition to the 3rd, the book took away the entire chapter 15, which is on Decision Theory. ... Read more

Book Description

This classic book continues to provide a foundation for mathematical literacy in business, economics, and the life and social sciences. Covers concepts ranging from introductory equations and functions through curve sketching, integration, and multivariable calculus. Helps readers connect concepts with the world around them through genuine applications, covering such diverse areas as business, economics, biology, medicine, sociology, psychology, ecology, statistics, earth science, and archaeology. Updates exercises, problems, and Mathematical Snapshots throughout. Improves writing style and mathematical derivations without sacrificing the book’s signature flavor. For anyone interested in learning more about introductory mathematical analysis.

Customer Reviews (2)

Nice book!
This book is really easy to understand. Its language is so simple. Anyone can read and understand it so well, even if your first language is not English. It gives nice examples with details to explain the solutions of the problems. It's about 19 chapters and they are: 0. Algebra Refresher. 1. Equations. 2. Applications of Equations and Inequalities. 3. Functions and Graphs. 4. Line, Parbolas, and Systems. 5. Exponential and Logarthimic Functions. 6. Matrix Algebra. 7. Linear Programming. 8. Mathematics of Finance. 9. Introduction of Probability and Statistics. 10. Additional Topics in Probablity. 11. Limits and Continuity. 12. Differentiation. 13. Additional Differentiation Topics 14. Curve Sketching. 15. Applications of Differentiation 16. Integration. 17. Methods of Applications of Integration. 18. Continuous Random Variables. 19. Multivariable Calculus. I recommend this book for students or readers for business because it teaches the basics of the topics (above)

You've got to have this!
The authors of this book seems to be a mathematician if you take a good look at all his other books. Great for intermediate mathematicians! ... Read more

Customer Reviews (87)

Fundamental
For all but the most gifted students, Rudin is a difficult book.If you are able to get through it, however, you have successfully taken one of the fundamental steps in learning advanced mathematics.While a good professor is the best aid for learning Rudin, other assistance can be got from:Shilov's Real and Complex Analysis (everyone studying math at the level of Rudin should have a copy of this classic (but cheap) Dover reprint), Kaplinsky's Set Theory and Metric Spaces (get it from the library) and Apostol's Mathematical Analysis (buy it used).

The Pinacle of Introductory Analysis
Walter Rudin's book barely needs introduction at this point.It has gained a reputation as the best text anywhere for an introduction to real analysis, and is the gold standard for many first year graduate courses in the subject.Rudin's work is a masterpiece of style and form, and his presentation is second to none.Care has been taken with every proof to make it as elegant as possible.The selection of problems typically ranges from those requiring a few minutes thought, to the fantastically difficult.

Therein does lie one of the two problems with this book, however.Occasionally Rudin relegates an important--and useful--result to the exercises where it could be overlooked by the unwary.There are some sections where more examples aimed at getting a student to practice applying fundamental concepts would be useful, instead of making them bend over backwards to find an answer.

The only other problem, which is often brought up as a criticism of the book, is that Rudin is often perhaps a bit too terse in his exposition between proofs.There isn't always a strong motivation given for a topic, which makes this book a difficult one to learn from without a good instructor.

Overall, it would be hard to do better than the so-called "baby" Rudin book.The price tag is a little steep for something so slender, but the content inside can easily outshine any other 3 similar texts in the area.This is an absolute must own for any aspiring analyst.

Solid and elegant
This book is well known for being terse.I will not refute this, but I will say that it is certainly not the tersest math book I have read (that honor might go to Samuel's algebraic number theory text).I am a graduate student in computer science, and I found this book to be enjoyable, well-structured, easy-to-read, and with excellent exercises.I probably would not attempt to use it for independent reading, though.The book develops calculus from the beginning, wasting no time, and giving almost no examples (unless you take the time to work through all the exercises).It is thus important to have a good instructor who can fill in the gaps as you go along.

The price, however, is ridiculous.

Simply the best ...
If you didn't use this classic in your first pass at elementary
analysis, you owe it to yourself to find a copy and work
through as many exercises as possible ... especially if you
plan to go further to graduate level work in mathematics.
Other books that are increasingly used for this subject still
leave readers with a 'maturity/sophistication' gap relative to
more advanced texts in real analysis, etc.

A course at MIT based on this text is presented at the link
below, with suggested coverage, exercises, solutions, etc. -
[...]

Pretty Good
I want to start by listing some of my complaints. First of all, I do not think it is entirely appropriate for a student's first exposure to analysis. The majority of students would be better off taking a an honors calculus or advanced calculus course first, so they can learn how to prove things about the continuity of specific functions or convergence of series before they start proving things and functions and series in general (a look at, say, Bartle's book over summer break would probably work just as well, but my point remains the same). Second of all, the exercises are, for the most part, extremely challenging. While this is not a bad thing by any means, the book would probably benefit by having a few extra, easier problems. Third, it could use some pictures, not that many, but they would certainly help illustrate some of the ideas. Fourth, a few of the proofs are very difficult to understand because they are perhaps too concise (for example, the Cauchy-Schwarz inequality in the first Chapter). Finally, $140 is prohibitively expensive.

That having been said, PMA is considered the classic of its genre, and for good reason. It is extremely well-written, if a bit concise, and forces the reader to do a lot of thinking. While the problems are challenging, they are very non-trivial, and again, force the reader to do a lot of thinking. You may be noticing a theme here. This is an excellent book if the student has enough experience with mathematical thought, however, many others will be lost.

Personally, I prefer Pugh's book, which I think addresses all the shortcomings I listed above, but I certainly see why many would want to stick with the tried-and-true PMA. ... Read more

Book Description
In this new introduction to undergraduate real analysis the author takes a different approach from past presentations of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians such as Dieudonne, Littlewood, and Osserman. This book is based on the honors version of a course which the author has taught many times over the last 35 years at Berkeley. The book contains an excellent selection of more than 500 exercises. ... Read more

Customer Reviews (6)

What a breath of fresh air!
Every once in a while, a mathematics book comes along that gets it right.While most math departments are filled with irrelevant, arcane texts; every once in a while a text will appear that is genuinely fun to read.Casella/Berger's fantastic statistics text is one of these; Pugh's Analysis is another one.

I cannot overstate how much I enjoyed this book - no matter what book your math department is using for undergraduate analysis, I would recommend picking this one up and reading it on your own.

Improves on the classic
As a previous reviewer has noted, Walter Rudin's Principles of Mathematical Analysis is the standard textbook for a rigorous analysis course. Rudin's book is very good because of the level of rigour and abstraction, the bredth of material covered, the way it forces the reader to fill in the blanks, and because of the challenging exercises throughout. In my opinion, Pugh has managed to improve on the classic in every aspect.

First of all, he does not develop all the concepts in same order as Rudin - first he develops the real number system, a few basic things about Cauchy sequences, and then moves onto continuity. Then he goes into a lengthy chapter on topology, which, in my humble opinion, is where the book first outshines Rudin. He defines compactness in terms of the convergence of subsequences, which is much more natural than the covering definition. He later proves that the two conditions are equivalent. In the third chapter, he develops differentiation and integration, much in the way Rudin does. In the fourth chapter, develops series and sequences (of functions). In the fifth chapter, he develops multivariable calculus, and the in the sixth chapter, he develops measure theory and the Lebesgue integral. Since there are fewer chapters than there are in Rudin's book, I think he develops the subject matter in a more natural, cohesive manner.

Rudin's book is excellent through the series and sequences of function. It is generally agreed that the book tails off after the seventh chapter, that is, he does not do as good a job with multivariable calculus and Lebesgue Theory. Pugh manages to do a good job throughout, so in addition to having a better chapter in topology, he is better than Rudin in those areas. I also believe that his treatment of series and sequences of functions is more interesting: Rudin treats them, for the most part, as distinct mathematical objects, and only briefly makes reference to the space of functions, whereas Pugh centers the chapter around the idea of function spaces (the heart of real analysis, really). Furthermore, Pugh uses illustrations (not too many, but enough) to illustrate certain concepts, and in fact, to simplify certain proofs. He also emphasizes the utility of geometric thinking in developing proofs, something which Rudin does not do. Furthermore, Rudin is notoriously terse; I think Pugh does a better job motivating and explaining the material without being "chatty" (the cardinal sin in mathematical exposition), while not insulting the reader's intelligence, that is, you are expected to fill in certain gaps on your own.

I would also like to emphasize the quality of the exercises in this book. There are many, many exercises - more than PMA, in fact. None of them are trivial. Many of them are quite challenging, on par with those in Rudin's book. Unlike Rudin, though, Pugh includes a fair share of easier, but still interesting exercises, which I think are essential for really getting a grasp on the material. He also has some problems, I think, which are a good bit harder than any of Rudin's, which is saying a lot, so there is something for everyone here.

Overall, I think this is the best book out there for an intro to analysis course. The price is also quite reasonable, considering how much math books tend to cost.

Would be better if solutions are provided
Great textbook, great afterchapter exercises! However, since the exercises are a bit challenging, it would have been better if solutions or hints for solutions are provided. By the way, I would be grateful if anyone could tell me where I can find solutions for the exercises.

Very good exposition, great problems
Real analysis is a genre with an established classic (Rudin) and a plethora of available books and resources. Unfortunately, most analysis books cost a great deal of money so the average reader will only purchase one or two texts. In evaluating which book(s) to purchase two questions should be asked:

1.) Why purchase this book rather than the classic of the genre?

2.) Is this book appropriate for me?

So why buy this book rather than Rudin? It has great exposition (as does Rudin), very well chosen problems (as does Rudin), but Pugh manages to improve on the standard by supplementing his written explanations with diagrams and pictures that Rudin mostly lacks. Additonally, the price stands at something less than half the cost of Rudin's book.

Who is this book appropriate for? This text delves into the topological underpinnings of analysis. It is not an "analysis-lite" textbook a la Ken Ross's Elementary Analysis. It is a rigorous treatment of the subject, and it has a comprehensive feel to it, covering topics like Lebesgue measure and integration, and multivariable analysis in addition to the normal topics one would expect. In short, it is appropriate for somebody who is seeking the challenges and rewards of a full treatment of what for many is a difficult subject.

It is a very good book that does not shy away from difficult material that no amount of explanation or good writing will make easy to learn, but of all the analysis books I've seen, this comes the closest.

A thorough text for an advance undergraduate
Having taken Pugh's honors analysis course, in which he used this book, I can strongly reccommend it to any student interested in the subject of analysis, especially students seeking to learn more than the average introductory real analysis book contains. Pugh's book contains advanced theorems and topics not often found in undergraduate level texts. Additionally, the problems are well thought out and tend to be of a high level of difficulty. ... Read more

Book Description
This is a textbook containing more than enough material for a year-long course in analysis at the advanced undergraduate or beginning graduate level. The book begins with a brief discussion of sets and mappings, describes the real number field, and proceeds to a treatment of real-valued functions of a real variable. Separate chapters are devoted to the ideas of convergent sequences and series, continuous functions, differentiation, and the Riemann integral. The middle chapters cover general topology and a miscellany of applications: the Weierstrass and Stone-Weierstrass approximation theorems, the existence of geodesics in compact metric spaces, elements of Fourier analysis, and the Weyl equidistribution theorem. Next comes a discussion of differentiation of vector-valued functions of several real variables, followed by a brief treatment of measure and integration (in a general setting, but with emphasis on Lebesgue theory in Euclidean space). The final part of the book deals with manifolds, differential forms, and Stokes' theorem, which is applied to prove Brouwer's fixed point theorem and to derive the basic properties of harmonic functions, such as the Dirichlet principle. ... Read more

Customer Reviews (9)

okay book
I taught a year analysis with this book. The book contains two semesters at least of undergraduate abstract analysis. The first half or so covers a first semester course with the usual things, metric spaces, continuity, sequences and series, and so forth. The second half or so has Lebesgue integration and multivariable analysis. I found it difficult to find another book having both semesters content in one volume, except for Rudin, which I chose not to use due to its price. So it fits a certain need. But I probably will not use this textbook again in my courses. I found the text to be way to concise, which might have been a result of the effort to fit so much into one volume. This is especially hard for the beginning half of the book, since the transition from calculus to abstract analysis tends to be conceptually challenging for students. Rudin is similar in some respect, since it is also concise. However personally I found it very natural to "fill in the blanks" in sketched proofs in Rudin, whereas in Browder I was more likely to develop a headache doing the same. Also I don't like the problems in the text. There's not really that many, and they tend to be multipart, too difficult and time consuming. The book could be improved by increasing the number of shorter exercises.

Concise, rigorous text
This book is for serious students of mathematics. A certain amount of mathematical maturity is needed in order to, fully, appreciate this book. It is also necessary to work through the examples in order to derive the greatest value from this text. Having studied, mathematics, over various years, from texts by Hardy, Rudin, Royden etc.,I feel that this book is a worthwhile addition to the armamentarium of a serious student who is interested in learning the tools of rigorous analysis.

Best Selection of Topics
I've read a few books on Real Analysis.Some attempt to cover too much, some don't cover enough.This book seems to include all of the essential topics without going overboard.It is also very easy to navigate.

Way too dense
The entire book is written in thm. prop. lemma. def. etc... form, with few breaks from this sequence of statements and proofs.While this seems to be the preferred style for a lot of textbooks, frankly I found it boring to read, and difficult to understand.That being said, the book covers a wide variety of topics, and goes into a good level of depth on each.The material is definitely worthwhile, but Browder seems to share the all too common mathematics teachers' curse of poor communications skills.

content OK, but problems with the typo density...
(...) First let me remark that talking about content, the book is very good. It contains elegance, rigor and the explanations seem OK. The problem however with the book is the typo density of most of the proofs. Some long proofs are presented as one continuous block without even no linefeed in between the facts. This makes some parts of the book rather uncomfortable to study from. I can 't stop starting to read these beautifull theory, but mostly, after one hour I give up, because "my eyes start dancing" ....
Tip for the author and editor : take exactly the same text, but spread it over 600 pages instead of 350 pages. This will make the book more expensive, but I will definitely buy it.. ... Read more

Book Description

Customer Reviews (4)

great for self-teaching
I am currently reading this book so I can learn calculus the "right" way. My undergrad courses in advance calculus and complex variables (notice I say variables and not analysis) were written for engineers and science majors that needed to know math, but not at this level. I find the sequence of topics (sets and functions, real numbers, sequences, ...) extremely helpful for understanding the material. One topic leads to another in a very logical and progressive manner. The problems range from the very easy "one liners" to more complex problems. The book contains hints to solving them at the end of the book. Very nice for self-teaching.

my advice : buy this as a reference
The pro's : Very good, everything is explained in a clear way, starting from the beginning, no gaps left in the proofs, and the material is abstract enough to motivate math lovers...In fact every math undergraduate and graduate should have this book as a reference, this cannot be a problem when you see the price.

The con's : Dover always has cheap price editions. While there is definitely a market for this, let' s face it : these editions have some disadvantages :While the contents of this book are very well suited not only as a reference but also to learn the material, the dense layout is not so comfortable to learn from. In that sense, the authors deserve a better edition... Maybe a question to the Dover guys : is it possible to bring your excellent science books in two editions : the existing cheap editions, and another more comfortable edition : bigger size, more whitespace on each page, ....

One of the better math textbooks
A very good introduction to real analysis, with all the appropriate theorems and proofs presented in a well ordered and understandable fashion. I use it regularly in my own teaching and research.

Very good
Everybody seems to think that Rudin's Principles of Mathematical Analysis is the best for whatever reason, & I agree that it's good for reference after being exposed to the material. Pfaffenberger doesn't construct the real numbers using Dedekind cuts, he makes a list of 13 axioms that basically say that the reals is the only complete ordered field. I think I liked this approach better than Rudin's (or Hardy's) more abstract approach. He also spends much more time developing metric spaces (including the Baire Category Theorem & nowhere dense sets, etc, which Rudin omits except for a couple exercises) and the Riemann-Stieltjes Integral. Then there's a short chapter where transcendental functions exp, sin & cos are defined which I think Rudin skips, and then introduction to inner-product spaces. Fourier series is introduced in the chapter on general inner-iroduct spaces (with a first look at Banach Spaces as an aside) rather than the chapter on sequences & series of functions. I also liked this better than Rudin's text or other calculus texts. Rudin includes a whole chapter on functions of several variables, but Pfaffenberger doesn't have anything on them. Instead, there's a chapter on normed linear spaces and the Riesz Representation Theorem, and then similar to Rudin, a chapter on the Lesbesgue Integral. This book has many more problems than Rudin's or Apostol's, and in general they are a bit easier. Of course every section has its difficult ones but the first ones are almost always just "verify that blah blah is true". The last ones are about on the same level as the ones in Rudin's book. ... Read more

Customer Reviews (17)

Excellent Intermediate Real Analysis Text
"Mathematical Analysis (2nd Ed.)," by Tom Apostol, does an excellent job of bridging the gap between standard introductory calculus texts and full-fledged treatments of topics in analysis.Apostol's book covers significantly more material than the gold standard of such texts, "Principles of Mathematical Analysis" by Rudin, and does so in a very different style.Where Rudin is brief and elegant, Apostol is thorough, detailed and friendly.Both Apostol's and Rudin's books have been around a long time, for very good reasons.

Unlike some intermediate texts, Apostol's book spends little time restating the particular results of elementary calculus (e.g., the derivative of sin x or x^n) in the new language of a more theoretical approach.Unlike Rudin and similar texts, Apostol *does* give detailed proofs, with thorough explanations.As a result of this approach, Apostol's book is not particularly well-suited to serve as a reference work for use by more advanced students or by professionals -- it is strictly a vehicle, and a very good vehicle indeed, for moving from elementary calculus to an introductory careful theoretical treatment of the material.Apostol does a particularly good job of presenting the "backbone ideas" of limits and continuity in a brief but very clear chapter (Chapter 4).

Apostol's problems are excellent and should be considered an important part of his presentation of the material.(This is one area in which Apostol perhaps surpasses Rudin, although MIT's online materials contain answers to so many of Rudin's problems that they now must be viewed as "worked-out examples!")Students find Apostol's tone, and the hints given in connection with the problems, to be helpful and engaging.

I suspect that the final few chapters of Apostol's book are used only rarely, due to the typical two-semester structure of real analysis courses (with a third semester being devoted to complex analysis).If true, this is a shame, because Apostol does a nice job of moving from a fairly standard treatment of the Lebesgue integral to Fourier integrals, multiple Riemann integrals and multiple Lebesgue integrals.

I should mention, as a minor point, that students can become confused, at least momentarily and episodically, by Apostol's parallel system of numbering (i) subsections and (ii) theorems and definitions.For example, the first line of page 166 reads "7.23RIEMANN-STIELTJES INTEGRALS DEPENDING ON A PARAMETER" and the very next line reads (in italics) "Theorem 7.38Let f be continuous at each point (x,y) of a rectangle . . . "Although the fonts differentiate these two parallel numbering systems, confusion can occur.

Great "second" book in introductory analysis...
This is an outstanding textbook that is also one of the more comprehensive books as advanced calculus and introductory analysis texts go.It makes an excellent reference because it is quite comprehensive, covering a number of topics that don't make it into most introductory analysis books.

Other reviewers have said enough about the quality of this book; I just want to add a few comments.The second edition of this book is very different from the first--it cuts out much of the material on vector calculus, but it adds material on Lebesgue integration, which it presents without the use of measure theory.

Anyone who finds this text a little too difficult might want to look at the book "Advanced Calculus" by Taylor & Mann.It moves a little bit slower than this book, is a little bit less abstract, and covers less material.This book is in some ways a logical "next step" after that book.I strongly prefer this book to the "baby" Rudin, both as a learning text and a reference.This book is more detailed, and the dependency of the material is less strict--it's easier to open this book to a specific topic and understand it without having to cross-reference earlier theorems.

One of the best I own...
I own books on mathematical analysis by Browder (0387946144), Douglas S Bridges (0387982396
), Haaser Sullivan (0486665097), Pfaffenberger(0486421740), Dudley (0521007542),Abbot(0387950605) and Apostol.

All books cover abstract multivariable spaces, except Abbott who limits himself to the real line.
None of these books are perfect, but of all these books Apostol is the one I prefer for the following reasons :

1. The contents :I think a beginning analysis course should serve two aims :
a. teach basic techniques that can be used in other theoretical oriented courses like physics,economics,...
b. at the same time let the students discover the beauty of abstract and rigorous math.

In this context Apostol has reached the ideal mix between abstraction and usability. He covers practical topics , used as a basis in a lot of other courses, but he does this by making the needed level of abstraction in order to proof everything in a rigorous way.

Each book is self contained, though none of these books give a good introduction into basic mathematical logic. However an introduction to set theory is explained well in all books.
Dudley 's beautifull book is the most abstract but requires the highest level of mathematical maturity.

2 Layout : The books of Haaser Sullivan , Pfaffenberger cover excellent material in a very clear way but they are cheap Dover editions, putting as much text as possible on one page. Browder 's contents I like most (and contains really excellent explanations), but his layout is also very dense and not always comfortable to read. The layout of Apostol is the best of all these books, its pages are well filled, but the difficult proofs contain enough whitspace for a confortable read.

3.Completeness and rigor : Apostol and all these books, except Abbott and Douglas S Bridges, proof everything they mention (exceptionally, they leaf a proof as an exercise, but then the proof is relatively easy enough if you understand the material). This is an approach I like : present the complete theory and then (like all of them do) create challenging exercises seperate from the basic theory.
In contrast, the book of Douglas S Bridges represents all material as one big exercise.This is nice if you have anough time, but most of us do not have that much time,I am afraid. Also Abbott has a lot of difficult proofs left as an exercise to the reader. But at the same time, Abbott is the best in motivating the reader. Abbott often provides excellent background in order to motivate the reader and sharpen the readers mathematical intuition.

While Apostol is not best on all the criteria mentioned above, Apostol scores good on all off them and as a consequence he has the best total average. This being said, I must omit that reading Apostol requires patience. Yes his explanations are clear, but can be very terse (especially his examples). Though, in principle everything is explained without gaps. This book requires reading every word carefully and take the time to reflect, but maybe that is the only way to learn advanced math.

Finally a remark about the price, I bought this book in Europe where it is much cheaper (check amazon.co.uk)

So compared with the others this a very good book.

The Cat's Meow
As stated by prior reveiwers, this books does assume that the reader is Mathematically mature (a saying most young Mathematicians despise), in the sense that he/she must be able to follow the logical development of any given arguement, be able to 'see' where and how topics are related as well as fill in any blanks that may present themsevles in a given definition/proof.Apostol, as compared to Rudin, does a nice job of filling in these blanks by adequately providing all of the necessary details within a proof.This book will provide the willing student with a solid foundation in elementary analysis as well as the confidence to persue higher analysis.The only draw back to Apostols book, aside from cost, is that the constant Theorem - Proof - Theorem format can be overwhelming at times and cause some readers to cover material too quickly.Despite the book's cost I would highly recommend this book over "baby" Rudin (that is, Principles of Mathematical Analysis) since Rudin is notorious for not filling in the blanks within a given proof and instead provides seemingly 'slick proofs'.

A cut above the rest...
I am currently studying from Apostol's book, completeing a year-long course with his treatment of the Lebesgue integral. While my experience with comperable analysis texts is not exhaustive, I am familiar with the more notable: "Baby" Rudin, Marsden,... So, I can confidently say that Apostol's text is among best covering the subject. His treatment is well modivated with examples, and his proofs, while not as not as "elegant" as those of Rudin, are surely more pedagogical in nature. Apostol has included a large amount of exercises that range througout the gamut of difficulty, and the material is peppered with a treatment of complex varaibles. Also, the readability is something to be attained by all authors of mathematics texts.

One drawback to the text is a too abstract approach to the Implict and Inverse Function Theorems. I found these to be the most challenging in the text, and I was forced to return to my copy of Stewart's Calculus text to re-acquiant myself with each concept. Also, at times Apostol falls into the pattern of Definition, Theorem, Definition, Theorem,..., but this seems to be only in the cases when ample preparation is needed to provide noteworthy examples; eg. Lebesgue integration.

So, in spite of the cost, I highly recommend this text for the study of real analysis (even for self study), although at [this price] there are bound to be others that have a higher value to cost ratio. Having completed the text (almost), I feel prepared to begin a more abstract study of analysis. ... Read more

Customer Reviews (1)

Explains Odd Number Questions, Skips Evens
This book DOES NOT have the answers to the even numbers. If you look in your original textbook (Introductory Mathematical Analysis), the back of the book just gives you the ODD answers without an explanation.

While this Student Solutions Manual does an OK job of explaining how they got an answer, I was under the impression that it did all the problems.

THIS BOOK ONLY SOLVES THE ODDS -- NOT THE EVENS. ... Read more

Book Description
Mathematical Analysis and Applications is a first introduction to Higher Mathematics for college students. Starting with a discussion of Real Numbers and Functions, the text introduces standard topics of Differential and Integral Calculus together with their Applications such as Differential Equations, Numerical Analysis, Approximation Methods. The subject is developed in an integrated manner, bringing out its essential unity and the inter-relationship between these topics. The Text is written in an informal manner and in lively language, but without sacrificing rigour in thinking and precision in writing, so essential in Mathematics. The book is divided in four Parts and seventeen Chapters, described below. A brief Historical Note is included at the end of each Part that describes the contributions of the mathematicians mentioned in the Text. The Book is ideally suited for class-room teaching and also for self-study to undergraduate students of Mathematics and of other disciplines such as Statistics, Physics, Computer Sciences or Engineering that use Mathematical Methods extensively. It is also a handy reference to teachers of Mathematics. ... Read more

Book Description
This is a logically self-contained introduction to analysis, suitable for students who have had two years of calculus. The book centers around those properties that have to do with uniform convergence and uniform limits in the context of differentiation and integration. Topics discussed include the classical test for convergence of series, Fourier series, polynomial approximation, the Poisson kernel, the construction of harmonic functions on the disc, ordinary differential equation, curve integrals, derivatives in vector spaces, multiple integrals, and others. One of the author's main concerns is to achieve a balance between concrete examples and general theorems, augmented by a variety of interesting exercises.Some new material has been added in this second edition, for example: a new chapter on the global version of integration of locally integrable vector fields; a brief discussion of L1-Cauchy sequences, introducing students to the Lebesgue integral; more material on Dirac sequences and families, including a section on the heat kernel; a more systematic discussion of orders of magnitude; and a number of new exercises. ... Read more

Customer Reviews (5)

Abeauty
I learned this material years ago but when I need to
look things up, this book has everything, well written.
For example the Taylor expansion with multinomial notation
is given here. The writer treats the undergraduate student's
intellect with respect. This is a serious book.

Just like every other Lang text
This book typifies Lang's style. If you enjoyed any of his other books you'll enjoy this too. Like seemingly all of his texts, it has a section on the inverse function theorem and makes quite a deal out of it. Overall, it is quite comprehensive, but there's little motivation for the proofs so things can be a bit boring. Both Rudin and Browder cover the same amout of material in far fewer pages, and they have better excerises too.

Solid, complete reference to basic analysis topics
Serge Lang's "Undergraduate Analysis" offers an impeccable selection of topics and exercises for the student wishing to broaden his/her knowledge of analysis. The proofs of theorems can be terse at times, but a hardworking student will gain much through a thorough reading of the text. Also, Lang concentrates many of his exercises on estimates, which is an art form that is slowly dying among undergraduates (and graduate students as well, sad to say). Many of his problems require only the triangle inequality (the basic tool of estimation) and ingenuity (and hard work) from the student. I would strongly recommend this text for anyone who wishes to fully understand and appreciate the results and techniques of basic real analysis.

okay
I personally don't care much for this book.It's too terse, and there are nowhere near enough examples.I have about 6 analysis books and this is the one I look in the least.It seems to cover a lot of stuff, but maybe too much-- it wouldve been better to focus more on some more elementary topics.For instance, he spends about one and a half pages introducing the derivative.So if you want a book that glosses over more elementary concepts and leans heavily toward a graduate level, this book's for you.At my school we were supposed to learn advanced calculus from this and it is not good for that at all.For advanced calc try robert stritchart's Way of Analysis (the best book on analysis I've ever read) and for analysis Intro to real analysis by kolmogrov (only 10 bucks or so and actually better than most books costing a 100)

Great intro to real analysis with logical format
Lang's book is an excellent introduction to real analysis that tries to build the reader's knowledge from the ground up.Lang starts with fundamental ideas from calculus and then proceeds in a logical way.Havingtaken Lang's course, for which this book was designed, it is clear to methat this book is a result of Lang's many years of experience teachingundergrads.I highly recommend this book to anyone who wants a solidfoundation in analysis. ... Read more

Book Description
Mathematical Analysis (often called Advanced Calculus) is generally found by students to be one of their hardest courses in Mathematics. This text uses the so-called sequential approach to continuity, differentiability and integration to make it easier to understand the subject.Topics that are generally glossed over in the standard Calculus courses are given careful study here. For example, what exactly is a 'continuous' function? And how exactly can one give a careful definition of 'integral'? The latter question is often one of the mysterious points in a Calculus course - and it is quite difficult to give a rigorous treatment of integration! The text has a large number of diagrams and helpful margin notes; and uses many graded examples and exercises, often with complete solutions, to guide students through the tricky points. It is suitable for self-study or use in parallel with a standard University course on the subject. ... Read more

Customer Reviews (1)

Excellent first intro to Analysis
This book pretty much copies the Real Analysis part of the Open University's excellent course 'Into to Pure Mathematics' which also covered Group Theory and Linear Algebra . If you're new to Analysis I cannot more highly recommend another book, though I would also recommend Spivak's Calculus which is at the same level, and pretty much covers the same material. I did the course (and have the book) and found it a breeze, all due to the excellently presented material, and the many examples, where like the SUMS books (of which I recommend most), all answers are givern in detail; which is important when you start more advanced maths (sic) and especially when you're teaching yourself!However it is only an introduction, but will prepare you for more advanced Analysis books like Rudin, Royden and my new favourite Knapp and his 2 Real Analysis books. Personally I wish they would produce more of the OU Maths courses in book format, as the material that I have read and studied has rarely been bettered in the standard books. Maybe this is a start of a new trend with the OU, I hope! ... Read more

Book Description
MATHEMATICAL ANALYSIS FOR ECONOMISTS BY R. G. D. ALLEN The general science of mathematics is concerned with the investigation of patterns of connectedness, in abstrac tion from the particular relata and the particular modes of connection. ALFRED NORTH WHITEHEAD, Adventures of Ideas To connect elements in laws according to some logical or mathematical pattern is the ultimate ideal of science. MORRIS R. COHEN, Reason and Nature MAQMILLAN AND CO., LIMITED ST. MARTINS STREET, LONDON 1938 PRINTED IN GREAT I3KITAIN FOREWORD TEUS book, which is based on a series of lectures given at the London School of Economics annually since 1931, aims at providing a course of pure mathematics developed in the directions most useful to students of economics. At each stage the mathematical methods described are used in the elucidation of problems of economic theory. Illustrative examples are added to all chapters and it is hoped that the reader, in solving them, will become familiar with the mathematical tools and with their applications to concrete economic problems. The method of treatment rules out any attempt at a systematic development of mathematical economic theory but the essentials of such a theory are to be found either in the text or in the examples. I hope that the book will be useful to readers of different types. The earlier chapters are intended primarily for the student with no mathematical equipment other than that obtained, possibly many years ago, from a matriculation course. Such a student may need to accustom himself to the application of the elementary methods before proceeding to the more powerful processes described in the later chapters. The more advanced reader may use the early sections for purposes of revision and pass on quickly to the later work. The experienced mathematical economist may find the book as a whole of service for reference and discover new points in some of the chapters. I have received helpful advice and criticism from many mathe maticians and economists. I am particularly indebted to Professor A. L. Bowley and to Dr. J. Marschak and the book includes numerous modifications made as a result of their suggestions on reading the original manuscript. I am also indebted to Mr. G. J. Nash who has read the proofs and has detected a number of slips in my construction of the examples. R. G. D. ALLEN THE LONDON SCHOOL OF ECONOMICS October, 1937 CONTENTS CHAP. PAGE FOREWORD ----------v A SHORT BIBLIOGRAPHY - ..... xiv THE USE OF GREEK LETTERS IN MATHEMATICAL ANALYSIS - - ...... xvi I. NUMBERS AND VARIABLES -------1 1.1 Introduction ---------1 1.2 Numbers of various types ------3 1.3 The real number system -------6 1.4 Continuous and discontinuous variables ... - 7 1.5 Quantities and their measurement ..... 9 1.0 Units of measurement - - - - - - - 13 1.7 Derived quantities - - - - - - - - 14 1.8 The location of points in space - - - - - 1G 1.9 Va viable points and their co-ordinates 20 EXAMPLES 1 The measurement of quantities graphical methods ---------23 . JpOJ ACTIONS AND THEIR DIAGRAMMATIC REPRESENTATION 28 2.1 Definition and examples of functions 28 2.2 The graphs of functions - - - - - - - 32 2.3 Functions and curves - - - - - - - 3 5 2.4 Classification of functions - - - - - - 38 2.5 Function types - - - - - - - - 41 2.6 The symbolic representation of functions of any form - 45 2.7 The diagrammatic method - - - - - - 48 2.8 The solution of equations in one variable 50 2.9 Simultaneous equations in two variables 54 EXAMPLES II Functions and graphs the solutionjof equa- tions ......... 57 III. ELEMENTARY ANALYTICAL GEOMETRY 61 3.1 Introduction ......... 61 3.2 The gradient of a straight line ..... 03 3.3 The equation of a straight line - - - 66 viii CONTENTS CHAP. 3.4 The parabola 09 3.5 The rectangular hyperbola - - - - - - 72 3.6 The circle 75 3.7 Curve classes and curve systems . - ... 76 3.8 An economic problem in analytical geometry 80 EXAMPLES III--The straight line curves and curve systems 82 IV... ... Read more

Book Description
We learn by doing. We learn mathematics by doing problems. This book is the first volume of a series of books of problems in mathematical analysis. It is mainly intended for students studying the basic principles of analysis. However, given its organization, level, and selection of problems, it would also be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. The volume is also suitable for self-study.

Each section of the book begins with relatively simple exercises, yet may also contain quite challenging problems. Very often several consecutive exercises are concerned with different aspects of one mathematical problem or theorem. This presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research. The collection of problems in the book is also intended to help teachers who wish to incorporate the problems into lectures. Solutions for all the problems are provided.

The book covers three topics: real numbers, sequences, and series, and is divided into two parts: exercises and/or problems, and solutions. Specific topics covered in this volume include the following: basic properties of real numbers, continued fractions, monotonic sequences, limits of sequences, Stolz's theorem, summation of series, tests for convergence, double series, arrangement of series, Cauchy product, and infinite products. ... Read more

Customer Reviews (2)

You only need two problems books: this is one!!
In your undergrad math career or maybe early grad, you only need two books for the subject analysis. These two volume book is one. Contains many challenging problems that will satisfy your ambitious Putnam apetite.

While the other one is, I will say, Problems in Mathematical Analysis by Berman ASIN: B0007AL4WG. This book contains tons of problems with little overlapping. It's comprehensive and collective! Should be called a comprehensive collection of problems in mathematical analysis (at undergraduate level). It's theeee most comprehensive one out there. While Kazac and Nowak's books Problems in Mathematical Analysis have more hard problems (Putnam type), this one helps you build up the knowledge you need to understand advanced graduate level Real Analysis. Highly recommended for econ/math/physics majors.

nice book of problems
contains a lot of examples and problems; maths can't just be learnt out of textbooks; you need this kind of book in order to go face to face with classic and sometime weird material as you can find there; very useful to set up exercices and tests when you teach this kind of things too... ... Read more

Book Description
We learn by doing. We learn mathematics by doing problems. And we learn more mathematics by doing more problems. This is the sequel to Problems in Mathematical Analysis I (Volume 4 in the Student Mathematical Library series). If you want to hone your understanding of continuous and differentiable functions, this book contains hundreds of problems to help you do so. The emphasis here is on real functions of a single variable. Topics include: continuous functions, the intermediate value property, uniform continuity, mean value theorems, Taylors formula, convex functions, sequences and series of functions.

The book is mainly geared toward students studying the basic principles of analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. It is also suitable for self-study. The presentation of the material is designed to help student comprehension, to encourage them to ask their own questions, and to start research. The collection of problems will also help teachers who wish to incorporate problems into their lectures. The problems are grouped into sections according to the methods of solution. Solutions for the problems are provided.

This is the sequel to Problems in Mathematical Analysis I (Volume 4 in the Student Mathematical Library series). Also available from the AMS is Problems in Analysis III. ... Read more

Customer Reviews (1)

You need ONLY two problem books in Real Analysis
In your undergrad math career or maybe early grad, you only need two books for the subject analysis. These two volume book is one. Contains many challenging problems that will satisfy your ambitious Putnam apetite.

THE KING OF ALL MATHEMATICAL PROBLEMS!!!!!!!!!

Problems in Mathematical Analysis (Hardcover)
by g. yankovsky (Translator), B. Demidovich (Author
Publisher: mir publisher; 4th Printing edition (1976)
ASIN: B000GTC2GA Should be called a comprehensive collection of problems in mathematical analysis (at undergraduate level). It's theeee most comprehensive one out there.



While Kazac and Nowak's books Problems in Mathematical Analysis have more hard problems (Putnam type), this one helps you build up the knowledge you need to understand advanced graduate level Real Analysis. Highly recommended for econ/math/physics majors.
... Read more

Book Description

Functions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.

Customer Reviews (2)

Much better than Rudin's "classic"!
Although Rudin's same-title book has a fame of Classical, I don't agree. Rudin's book may fit students majoring in Mathematics. Students from any other specialty will find Rudin's book very unreadable. I just wonder why Rudin got a fame from such a bad-written book. Personally, I will prefer Parzynski's much clearer and simpler style. It is simple, but complex enough to let you learn most skills required for Calculus. After carefully studying this book, you can go into Real Analysis and Complex Analysis.

A good "pre-Rudin" book
Anyone taking mathematical analysis at a sophisticated level,
e.g. using Rudin's "Mathematical Analysis", would benefit from
this book which pays much attention to continuity and differentiability in a more concrete fashion than one typically encounters in a more advanced book. The first few chapters on sets, intervals, compactness, and connectedness may seem daunting, but the book becomes much more friendlier after that. ... Read more