Reviews (4)

Decent book, if you can get it cheap
I strongly urge any serious math student to own a copy of both Rudin's Principles ("Baby Rudin") and his Real and Complex Analysis ("Adult Rudin").The former is absolutely essential- without completely mastering continuity and convergence on the basic metric space topology on R^n, higher math is going to be quite a pain.The second is good because it puts the major ideas of basic analysis- Radon measures, L^p spaces, rudiments of Hilbert and Banach Spaces, differentiation and integration, Fourier and Harmonic Analysis, Holomorphic and meromorphic functions, etc. all in one nice volume, although the problems may be too challenging or tangential to master the material by doing them.

With that said, I don't like this book as much.Perhaps because the problems don't provide great movitation for the theorems- in any event, I would recommend using at least two books to understand functional analysis.One that emphasizes a rigorous approach to the theory involved, and another more applied book that allows you to play with the new tools to solve the problems functional analysis was invented to solve; quantum mechanics, for example.

Reed and Simon is a good book, although I'm sure physicists or physics students would probably complain about it for the same reason I like it- its very mathematically rigorous and has a ton of problems- 30 to 60 on average at the end of each chapter, with only a few digressions into applications into quantum physics or elementary QFT.Get this with some Springer text, like Elements of Functional Analysis.

One more note- Rudin's book is broken up into three parts- one on TVS (Topological vector spaces) that combines topological properties of a space (for example, local convexity or local compactness) with the usual vector-space operations to set the spaces where operators act.

The second section deals with distributions- I regret that one failure of "Adult Rudin" was to emphasize the abstract integral as a linear functional, because this would have helped to make the concept of a distribution more clear.

While the introduction to distributions and their connections to Fourier analysis and differential equations is nice, the text gets bogged down with proofs about convolutions that are highly technical (and make either good practice or a good time for Rudin to actually use, for once, "The details are left to the reader...").

Finally, Rudin introduces operator theory, although it could go much more smoothly- the proofs come off as way too technical, a far cry from the "slickness" his proofs are often accused of being in the graduate analysis text.

All in all, there's some interesting problems to do, but you're not going to understand the applications of Functional Analysis to quantum mechanics or PDE (other than distributions a little), where other, more applied (read: easier) books may give nice problems about applications of Hilbert space methods, such as variational techniques or Fredholm theory.

Modern topics in math.
"Modern analysis" used to be a popular name for the subject of this lovely book. It is as important as ever, but perhaps less "modern". The subject of functional analysis, while fundamental and central in the landscape of mathematics, really started with seminal theorems due to Banach, Hilbert, von Neumann, Herglotz, Hausdorff, Friedrichs, Steinhouse,...and many other of, the perhaps less well known, founding fathers, in Central Europe (at the time), in the period between the two World Wars. In the beginning it generated awe in its ability
to provide elegant proofs of classical theorems that otherwise were thought to be both technical and difficult. The beautiful idea that makes it all clear as daylight: Wiener's theorem on absolutely convergent(AC) Fourier series of 1/f if you can divide, and if f has the AC Fourier series, is a case in point. The new subject gained from there because of its many sucess stories,- in proving new theorems, in unifying old ones, in offering a framework for quantum theory, for dynamical systems, and for partial differential equations. And offering a language that facilitated interdisiplinary work in science! The Journal of Functional Analysis, starting in the 1960ties, broadened the subject, reaching almost all branches of science, and finding functional analytic flavor in theories surprisingly far from the original roots of the subject. The topics in Rudin's book are inspired by harmonic analysis. The later part offers one of the most elegant compact treatment of the theory of operators in Hilbert space, I can think of. Its approach to unbounded operators is lovely.

Reviews (16)

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 the US, and around the world.

The book is divided in the two main parts, real and complex analysis. But in addition, it contains a good amount of functional and harmonic analysis; and a little operator theory.

I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know.

What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.

Some of the exercises are notoriously hard, but I think that is good: It simply means that they serve as work-projects when the students use the book. And this approach probably is more pedagogical as well.

After surviving some of the hard exercises in Rudin's Real and Complex, I think we learn things that stay with us for life; you will be "marked for life!"

Review by Palle Jorgensen, September 2004.

Welcome to the self-service analysis center!
This year we have been using Walter Rudin's treatise as the main text for a standard first-year graduate sequence on real analysis, backed up by Wheeden/Zygmund's book on Measure and Integral, and the two seem to complement each other quite nicely. Rudin writes in a very user-friendly yet concise manner, and at the same time he masterfully manages to maintain the high level of formality required from a graduate mathematics text. To be totally honest, a few years ago my very first attempt at learning graduate-level real analysis in a classroom setting (via Folland's book) was not successful, as I found the exposition in Folland very dense and rigid, and the homework problems too difficult to do. Rudin's book however is a lot more accessible for the beginning graduate students who may not have had any more than some basic exposure to measure theory in their upper division analysis classes. One point to keep in mind is that Rudin developes the measure in the more formal axiomatic way, instead of in the more concrete constructive approach. In the constructive approach, one first introduces the "subadditive" outer measure as a set function which is defined on the power set P(X) of a nonempty set X. One then proceeds by showing that the restriction of the domain of the outer measure to a smaller class of subsets of X (a sigma algebra M), obtained via applying the Caratheodory's criterion, results in a "countably additive" set function which is called a measure on (X,M). (The latter is the approach taken in both H.L. Royden and Wheden/Zygmund). The formal approach is not very intuitive and is less natural for a beginning graduate student who might not have developed a certain level of mathematical maturity yet.

Also, Rudin does not discuss some of the more advanced or interdisciplinary topics such as distribution theory (Sobolev spaces, weak derivatives, etc.) or applications of measure theory to the probability theory, both explored in the book by Folland. Last but not least, it's worth noting that contrary to the common practice, Folland includes many end-of-chapter notes where he outlines some important historical aspects of the development of the topics, and also gives a few references for further study. For example, in the notes section at the end of the chapter on Lebesgue integration, he mentions --and briefly outlines-- the basics of the theory of "gauge integration" (also called Henstock-Kurzweil theory) which serves to construct a more powerful integral than that of the Lebesgue's. As another instance, having already defined and used "nets" within the chapter on topology, in the end-notes Folland also introduces "filters" and "ultrafilters". These are all machineries which have been developed to play the role of the metric space sequences in general (locally Hausdorff) topological spaces, but for some historical reasons, ultrafilters have nowadays taken a backseat to the nets (the older general topology books usually prove the Tychonoff theorem using ultrafilters). All said, I can recommend taking up Royden as your very first approach to measure theory, then based on how well you think you have learned the first course, move on to either Rudin or Folland for a more advanced treatment. Please note that the other books I have mentioned above do not discuss complex analysis, a subject which is also masterfully presented in Rudin. There are however a few other equally well-written complex analysis books to pick from, for example John B. Conway's classic from the Springer-Verlag graduate series, or L.V. Ahlfors' masterpiece, to name just a couple.

A Comprehensive Guide to Analysis
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.

Reviews (73)

An excellent textbook
I think mathematics is a part of our culture.That's why, as a non-math major, I wandered into a very serious analysis class for mathematics majors.That might have been a disaster for me.Luckily, we used this book as a text, and it saved me.I read the whole book and diligently did all the exercises (of course, back then, it was the first edition, with only 227 pages and 140 exercises; it's somewhat more now).And that is my recommendation today.Read the book carefully and do as many exercises as you can.It certainly isn't easy.But it isn't, um, countably hard either.

The material in the book is self-contained.I guess that in theory, it could be mastered by any bright 14-year old who had learned some high school algebra and geometry.But I would surely recommend having much more mathematical sophistication than that as a prerequisite!

If you haven't learned the language of mathematics before, you'll enjoy the use of terms such as "countable," "real," "rational cuts," "measure," "ring," and "complete." By the end of the book, when the author claims that a proof (involving Cauchy sequences no less) is complete, you'll barely be able to suppress a desire to ask "Does every Cauchy sequence in the proof converge?"

In the first edition of this book, Rudin did mess up a little in his section on "the integral as a limit of sums." His theorem as stated was false.We cruelly dubbed it "Rudin's Last Theorem."Worse, he had used it "to prove some elementary properties of the Stieltjes integral."But that was all straightened out by the second edition.

I especially like the first couple of chapters.They give most readers the confidence to continue.And the final chapter, on Lebesgue integration, is very well written.One note of warning, though.Rudin begins this chapter by saying, "Some of the easier propositions are stated without proof.However the reader who has become familiar with the techniques used in the preceding chapters will certainly find no difficulty in supplying the missing steps."That is an exaggeration.It takes work.After all, this is, um, real mathematics you'll be doing!

I'm thankful that I was assigned this as my textbook.

A masterpiece
I absolutely agree with Professor Jorgensen.

I loved it when I was a student of physics, and I still love it because I tend to consider it as my personal standard in Classical Mathematical Analysis (and not only): sort of a "pacemaker" which sets the qualitative level to go back to just every time one is a little confused about what to do and where to go ;)

Reviews (17)

Standard Analysis Text; Necessary, but not Sufficient!
Very useful textbook, easy to follow.
A little over-simplified though.
Doesnot leave a lot of gaps, but then that doesnot leave a lot of room for imagination either.
At times it gets toooo wordy, if you know what I mean. Less ideas and more talk.
A stark contrast to Rudin, but can be very useful if both are used together. However, if you need to choose between those two, go with Rudin.

this book is just plain good.
I began as a graduate student in applied maths less than a year ago; all of the students that I spoke with prior to that said that real analysis with rudin's book was their worse & hardest class..
So when I walked into MTH 5111 Real Variables I thought oh *&^% what am I in for?? but then I picked up the Royden book and I understood the way he was presenting the materail.. the book is very stright to the point + leaves channelgning problems to the HW sets but the autor clearly outlines. I have learned more from this book and course than any other...

Reviews (5)

Frustrating
It seems that the higher up you go in the Mathematics curriculum, the poorer the books you meet.In my honest opinion, a book should help you learn and understand the material as quickly as possible.Otherwise, you might as well be given a list of definitions, stuck in a closed padded room and asked to come up with all the theorems by yourself.Unfortunately, there are too many graduate textbooks out there written by individuals who seem to have no desire to make the ideas they are trying to present as clear as possible.There's no educational philosophy.This book falls under that category.For example, this book is almost completely devoid of any examples.I don't know about you, but from example, is how I learn.I could go through this book much faster, if there were some decent examples.You can tell me a thousand times what a sigma algebra is, but if you don't give me some decent, worked-out examples which might tell me why tell me why I should learn it (other than because I'll fail the course), I'm going to forget the definition after 5 minutes.Secondly, it would help if there were more pictures.A picture is worth a thousand words.Third, some of the definitions are not worded as well as they should be.last night I spent ten minutes trying to figure out whether the definition for x-section Ex = {y in Y : (x,y) in E} meant that "for all x," or just "for some x?"It turned out it meant "for fixed x."But nowhere was that little tidbit of information written.Ten minutes may not sound like much, but if you have to read 10 pages before you get to pleasure of spending 10 hours with the homework problems, that translates into a lot of time you could spend doing other things if only this book were presented in a manner which would enable you to learn the material more efficiently.I give it two stars primarily because some of the homework problems aren't too bad.If you have a choice, have a look at Kolmogorov and Fomins book on Real analysis.It's not perfect, but the material in it is organized better.(It's not as DENSE)Plus it's a Dover book, and therefore much cheaper.

Could have been great
I speak as a graduate student in applied math.I really like this book but was bothered by its flaws.Nevertheless, with a good instructor, this text can make for a good learning experience.

Positives:The book is well organized.It builds in a reasonable way so that I could focus on the material in the book and develop my understanding as I went.The book is reasonably well contained.Outside of a reasonable level of basics (a BA or BS in math) the proofs and most of the problems use material developed earlier in the text.I found the book very interesting -- I especially liked the topics presented in the last few chapters.

Negatives:Lots of typos - the author's errata sheet is woefully incomplete.Too few expamples.Too condensed - sometimes to the point of incomprehensibility or even error.The contents of a whole course may be condensed in to a single chapter or even a single section.

Things to be aware of:You should be comfortable with advanced calculus, topology, set theory, and algebra (linear and modern).It also helps to have had some basic real analysis.I highly recommend that you've seen Fourier transforms, Dirac deltas (distributions), and continuous probability.You aren't going to learn these here - you're going to see how measure theory is applied to them.

Reviews (1)

I've seen a better book
I'm taking a course in Real Analysis and using this text. Several difficulties arise while reading this book. It's either because I'm a weak student, or the text itself. Here are some facts about the book:

1. Manytopics are covered.

2. A lot of important results have been left asexercises. And I really mean *many* !So, you need a guide (like ateacher).

Reviews (15)

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.

Reviews (1)

Reviews (16)

Lots of description... a bit too much
The book is excellent for self-study with its in-depth descriptions.The reader should look out for the occasional misprint (even in definitions, which makes it a bit more difficult), but if the book is read carefully these errors expose themselves.The reader should highlight definitions for later reference (since they are sometimes within the text).

In use for a class or for reference, this book is too wordy.Definitions are difficult to spot in many cases (some are written directly within the text).Descriptions also drag on too long in many cases, making it difficult to read the entire text.The book is by no means concise, which, after sitting in a lecture on the topic, makes much of the description too repetitive.

The problems the book offers are very good, in that they require thought but are also possible for someone fairly new to analysis (although this isn't unique to this book).The proofs are clear and many are quite elegant.They need no explanation other than what is in the text (how it should be).

One last comment on the book itself -- this book is too big to be a paperback and hold up under typical use.Look for this one in a hardcover edition, if you can find one.

Recommended instead of this book: "Principles of Mathematical Analysis," by Walter Rudin.This book is concise and clear and most appropriate when taken with a course, but must involve careful reading for self-study in comparison of Strichartz's book.

Very lucid and ideal material for learning real analysis
Most books on mathematics simply dump concepts,equations and examples and let you figure out what to do. Not this one. The book is written in a passionate manner where the author takes pains to explain why we are going in a particular direction and the goals. The style is extremely lucid and informal, something unusual for a subject that is steeped in formal mathematics. Yet the author presents, explains and covers all the formal theorems, concepts etc . The book also has excellent exercises. A truly noteworthy achievement. I would highly recommend this to anyone (especially self-study) trying to learn this subject.

Isbn: 0821820508
Sales Rank: 364648
Subjects:  1. Calculus    2. Mathematical analysis    3. Mathematics    4. Science/Mathematics   

Isbn: 0821820516
Sales Rank: 397339
Subjects:  1. Mathematical Analysis    2. Mathematics    3. Science/Mathematics    4. General   

Reviews (16)

why 4 stars and not five?
I was a (French)graduate student in France some 25 years ago and I would have been delighted to use this book if translated in French; I had to rely on Cartan's book which is a very good book too but which takes for granted that one already knows quite a lot on complex numbers, series, convergence and topology...As a substitute to Cartan, there was a translation of Rudin's real and complex analysis which begins with measure theory...Anyway, it is very difficult to learn this subject in any book without advice from instructors and attending lectures.
There could be more worked examples in this book but it is nota self teaching book (neither is Cartan's...which is very similar in essence to Ahlforsbut more narrow minded). For a more "basic" book in the subject, see Marsden's Basic complex analysis but proofs are often mixed up with exercises...which does not suit everybody. My final point is the following: this book contains much more stuff to work at or to think about than its French counterpart; moreover,in this book, efforts are made to avoid formalism (Bourbaki?). US maths students are very lucky indeed. But the book is certainly too expensive.

A Classic
Another classic text from graduate school (text for class taught by P.L. Duren) providing a background in introductory complex analysis.This book is nicely written with some elegant exploration of the motivations and backgound for a number of the central concepts.This may be surprising given the physical slimness of the text (I noticed elegance of the exposition and attention to motivation on a recent reread of some of the book after nearly twenty years -- I had not remembered this exposition, perhaps because the reading in graduate school was not quite as "liesurely" (unless "fear driven" and "pressured" are synonyms for "liesurely").The theory topics are nicely covered -- if, however, you are an engineer looking for methods of calculating complex intgral there are other texts.

Reviews (4)

sweet dude
I dont like lang's algebra, ugrad linear algebra, or diff/riemannian manifolds books all that much, but i LOVED this one.

I think an undergrad with calculus and patience can read it.
there are characteristic lang-style things like research-oriented material, and he actually has examples. He covers topics towards the end of the book which arent common elsewhere, so i've never put it down. I am not a mathematician and I like this book. It's in one of my standard 8 books that I dont leave home without (4 physics 4 math)

A good book, but not for beginners.
if you want an introduction to complex analysis, I advise you to pass onthis book, and read Churchill and Brown's introductory book. Having saidthis, part I of Lang's book will seem mostly review if you follow myadvice. Part II, on Geometric Function Theory, is more advance materialthat is presented reasonably well.

Not TOO complex
A person with absolutely no knowledge of complex numbers couldbegin with page one of this book.However, I think that some exposure to analysis is helpful before finishing the first chapter, but not necessary.I foundthis book easier to read & understand than some real analysis books,yet it helped me further understand real analysis in the process.I'm surethis is due to mere repetition of some of those concepts over a differentfield.As the author mentions in his foreword, the first half of the bookcan be used as an undergraduate text (Jr/Sn years) and the second half canalso, but I would NOT have enjoyed it in undergraduate studies.I found itworthy of a first course in complex numbers at the graduate level.Iespecially liked it after studying real numbers.The placement of thechapter subject matter can be altered (to some degree) to ones liking.Ithink Lang has provided good examples & problems.There's a solutionsmanual (by Rami Shakarchi) for this text somewhere.

A brief discriptionof the chapters (some of them at least):

Chp 1:basic definitions &operations, polar form, functions, limits, compact sets, differentiation,Cauchy-Riemann eqs, angles under holomorphic ("differentiable")maps.

Chp 2:formal & convergent power series, analytic functions,inverse & open mapping thms., local maximum modulus principle

Chp 3: connected sets, integrals over paths, primitives("antiderivatives"), local Cauchy thm, etc

Chp 4:windingnumbers, global Cauchy Thm, Artin's proof

Chp 5:Applications ofCauchy's integral formula, Laurent series

Chp 6:Calculus of residues,evaluation of complex definate integrals, Fourier transforms, etc(funstuff)

Chp 7: Comformal mapping, Schwarz lemma, analytic automorphisms ofthe Disc

Reviews (22)

Excellent intro. to complex analysis!
This course was my first exposure to the mathematical field of analysis at the undergraduate level, and our school ditched Gamelin's book used two years ago in favor of this book.Just to give you an idea of the difference a book makes (it was the same teacher for both courses, mind you):when Gamelin was used, EVERYONE dropped out of the course; when Brown/Churchill was used, only one person dropped the course and half the class received A's!

Truly, this is a remarkable shift, and this book had a lot to do with it.I thought the organization was flawless (note:you will have to go through the book in order, as many examples depend on previous material), and starting from the beginning with the definition of a complex number was definitely the way to go, as about 1/3 of my class had never seen a complex number before.I loved the fact that there were many examples worked out (never explicitly showing people how to do the end-of-section exercises, but showing them the methods for where to go) and the major theorems were alloted many pages for clear proofs with diagrams and detailed explanations (an entire section was devoted to a proof of the Cauchy-Goursat theorem!).Also, the choices of problems were superb, with some routine exercises meant to get you thinking along the right tracks followed by some very difficult ones.Basically, enough to challenge even the ablest math student, but enough for the average one to get a grasp on the concepts as well.

The book also provides an advantage for the instructor as to what applications to teach.Granted, chapters 1-6 cover almost all the theory, but 7-12 are all applications (7 is "usually" considered theoretical as well, but it is called "applications of residues!") in physics, advanced calculus and geometry, and engineering.So, a professor could choose to emphasize only the theoretical parts and save the apps. for independent study (which my prof. did) or could teach the relevant theories coupled with some of the applications (conformal mapping with fluid flow and heat flow, for example).It truly is a versatile book.

I noticed a complaint on here about not having enough examples or worked-out proofs.Well, to that individual (and any others who might be having the same problem), this book is meant for an upper-level undergraduate course, which means that there are going to be less examples worked out in great detail, the proofs may just be thumbnail sketches, and the problems will not have a quick reference page in the chapter for a formula or method like in calculus, for example; even though the book is versatile, a lot of the learning still falls on the student's shoulders.

My one and only gripe is that the book didn't take a lot of time to spell out how to perform a delta-epsilon proof for limits, which is one of the basic proofs in analysis.But, luckily, I had a very patient instructor who was willing to walk it through with me (most of the rest of the class had already had real analysis, so they didn't need to go over it).But, still, it's not enough to take it down a star, in my opinion.

They say this book is among the canon of undergraduate mathematics, and I can certainly see why.What a great introduction to complex analysis!This book will definitely be accompanying me to grad school!

excellent! (as an intro)
This book would be really good for a physics or engineering student, or for a math student as an intro since it isn't as theoretical as other books. Everything is explained very clearly, with lots of examples but still with some hard/interesting problems also. The first 7 chapters cover all the theory, which is the usual stuff like complex numbers, differentiability & Cauchy-Riemann equations, integration & Cauchy integral forumla/theorem, series, Taylor series, Laurent series, residues, poles, solving real integrals with residues, etc etc. Chapters 8-12 cover applications which I admit I don't know a lot about, but if the 1st half of the book is any indication, they are good also. The text covers everything in a CONCRETE way, as opposed to an abstract way using concepts from topology like connectedness, compactness, open/closed sets, star-shaped sets, etc. For that stuff I used GJO Jameson's Complex Functions text & I also like the one by Ahlfors & the one by Jerrold Marsden. I think every scientist or engineer would find this Brown/Churchill book very helpful. :-)

Wonderful intro to complex analysis.
I used this text for an undergrad course, and I thoroughly enjoyed it.The book, as the title suggests, is ideal for physicists, engineers, and applied mathematicians, in that it hits the important and powerful features of complex analysis which are tremendously useful for applied work.What's more, it succeeds in doing so without sacrificing mathematical rigor, and without getting bogged down on stodgy formalism.

As for the scope of the book, I believe it can be fairly stated that just about everything in the book should be studied and mastered by readers doing applied work.From my own experience, everything covered in the book has turned out to be relevant at one point or another.It can be said without exaggeration that this book is a gold mine.

Reviews (7)

the only book you need for complex analysis
that all i have to say about it.

Excellent book -- Unique selection of topics
This is the only textbook that I know that introduces and explains the Hilbert-Riemann problem in a pedagogical way. If anyone knows of any other such book, please tell us. It also deals in a very introductory way with all sort of really nice topics that one cannot find discussed (at a really introductory level) in any similar book: the Painleve property, the classification of singularities, asymptotic expansions, etc, etc. All very powerful applied mathematics.

A very good text!
A very good text!

The best description of this book is that it provides a comprehensive, classical treatment of the subject with a modern touch and serves ideally the needs of anyone studying Complex Analysis.

Starting from the foundations of defining a complex number, through to applications in the evaluation of integrals, the WKB method, Fourier transforms and Riemann-Hilbert problems, the book covers a lot of ground in an easy to follow style. The chapters are long, but logically broken down into digestible sections and interspersed with well illustrated diagrams, numerous worked examples and exercises.The end of chapter exercises provide further opportunity for reinforcing the methods and there's a useful section at the end giving brief hints and answers to selected problems.

Complex Variable analysis is treated from the definition of an analytic function and its relation to the Cauchy-Riemann equations, and in turn their application to an ideal fluid flow. The ideas of multi-valued functions, complex integration, and Cauchy's theorem are excellently treated, as are the consequences: the generalised Cauchy integral formula, the Max-Mod principle, and Liouiville and Morera's theorems.

The rest of the first part of this book, which is essentially pure mathematics, deals with Laurent series, singularities, analytic continuation, the Mittag-Leffler theorem, the ALL IMPORTANT Cauchy Residue Theorem, dealing with branch points, Rouche's theorem, and their application to Fourier transforms.

The second half starts off with perhaps the best I have seen on Conformal Mappings and their application to physical problems in Fluid Mechanics and Electromagnetism.Asymptotic evaluation of integrals covers methods like Watson's lemma, the method of steepest descent, and the WKB method.

A good combination of pure and applied mathematics, though the book avoids either the rigour of classical works such as Whittaker and Watson or the marvellously visual presentation of Tristan Needham.