Product Description
Excellent undergraduate-level text offers coverage of real numbers, sets, metric spaces, limits, continuous functions, series, the derivative, higher derivatives, the integral and more. Each chapter contains a problem set (hints and answers at the end), while a wealth of examples and applications are found throughout the text. Over 340 theorems fully proved. 1973 edition.
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Customer Reviews (9)

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

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

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

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

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

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

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

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

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

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

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

Product Description
Now available in paperback, this radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Although aimed at the complete beginner, professional mathematicians and physicists will also enjoy the fresh insights afforded by this unusual approach. ... Read more

Customer Reviews (43)

A fun math book!
This book is a real treat.It gives a beautiful visual description of complex differentiation.I was happy to learn how to visualize this due to my "lack of four-dimensional imagination".The author elevates the abstract symbolic mathematics into a living picture of vectors transforming through stretches and twists.He takes the reader into the kitchen and uses a knife and rolling pin to really illustrate conformal mapping and its relationship to complex differentiation.Using the author's words to summarize:"The basic philosophy of this book is that while it often takes more imagination and effort to find a picture than to do a calculation, the picture will always reward you by bringing you nearer to the Truth."If that isn't the Richard Feynman philosophy!

The best mathematics book I've ever read
This book lives true to its name. Most other complex analysis books put so much focus on the algebra of complex analysis that the geometry of complex numbers are almost forgotten. However, this book puts geometry first and foremost. In fact there is very little algebraic proofs at all throughout the book. Rather than going through complex and detailed algebraic proofs he mostly relies on the readers geometric intuition to guide the reader to the proper conclusions.

My favorite aspect of the book is with the authors writing style. I had to read this book for a graduate level math class and I was expecting a dry hard read. However, to my shock this read more like a philosophical discourse. Don't let this fool you, though. It is still very heavy stuff. The approach is just different (and fun).

Visual Complex Analysis
This text ended up being the only required reading for the undergraduate complex analysis course which I am taking this spring.I decided to pick it up early to review and its been nothing but a pleasure.I am a mathematics/biology major but definitely not naturally gifted by any means in regards to math, only through diligent work has Needham's "Visual Complex Analysis" made the plunge into this topic pleasant.I feel the author does a good job a using unique examples and geometric illustrations to connect proofs to more visual descriptive explanations.Everyone is bound to find something in this book that they will like.
The only thing I noticed is that the author chooses a style that leads to more theory and proofs then others. I do not mind this approach.There tends to be not as many "plug-chug" examples or problems to work through.Background in vector calculus, linear algebra, and differential equations has helped me in reading this book.Two thumbs up!

Illuminating
It is one of the most illuminating text on Complex Analysis. Visualization is missing in almost all theoretical texts on this subject. Compliment to the author.
Sincerely Yours
Aldo Materassi

Recomendado
A typical judgement about this book would be as follows; "works good on the concepts and intuition, specially the visual/geometrical, but lacks mathematical rigor". But, if mathematical rigor is (for you as for me) an intrinsic logical structure beneath the argument, this book is rigorous enough even for the common math snob (I mean i don't believe that the goal of a math proof, in a textbook, is to overcome the student's moral with the full logical-set theorist notation of that proof but to make clear why the proposition is true and to make the reader think about the result). Overall it's a joyful book to read, enlightening in every chapter, the order in the presentation of the content is smooth and, have to say, the writer's attitude is modest and patient. I would recommend it to anyone with the basic knowledge related to a standard calculus course, and think it would even de of use for a graduate student in mathematics or physics. ... Read more

Product Description
Abundance of worked-out examples and over 300 problems (some with hints and answers) make this an excellent text for one-year graduate or undergraduate course, independent study. Unabridged (1984) reissue of first Dover (1972) edition.
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Customer Reviews (8)

great Complex Analysis text
Great complex analysis text -- I will use this for teaching two semesters of complex analysis. Lots of examples and gives all the necessary theory. It explains even basic concepts in analysis and topology as needed, so the student doesn't have to look for other references. Since it is a bit old, in a very few places the terminology/notation is a little outdated. The book contains lots of exploring of the individual complex functions and at the same time all the theory I would ask for. In some places I would do the proofs a little differently -- the author is very hands-on.

Nice book on complex analysis
On 2001, I took a graduate course on complex analysis having very little previous knowledge about it, and this was the book I used to help my self. This book resulted being a delight, it's wonderfull the way it is written the clarity, all theorems with proofs and starts from the basics till more advanced topics. I recently read it completely once again and the same thing happened: is a wonderfull book, my plan now is to get the bigger work of Markushevich, to extend the knowledge. In one word buy it you won't regret it! and is cheap!(by the way, I am the same person as the one on the review called "Great book on complex analysis", the only difference is that on that occasion I didn't leave my name so now I've done it finally)

a classic
Silverman's book starts at complex numbers functions and sequences, and it covers some central aspects of complex function theory, elementary geometry, Mobius transformations, harmonic and analytic functions.

The central topics are (in this order) geometry of the plane, fundamentals of complex numbers, limits and a brief calculus review, calculus and geometry of the plane, harmonic functions, complex numbers, integrals, power series and analytic functions, and the standard Cauchy-and residue theorems, ending with a brief chapter on conformal mappings.

The book was published first in 1967, but reprinted since by Dover. It is suitable as a text or as a supplement in a standard course in complex function theory, at the late undergraduate level. While it contains the standard elements in such a course, we note that a systematic treatment of power series comes relatively late, in Chapter 10, beginning on page 195 (halfway into the book.) Some readers might want to begin with that.

Of other Dover titles on the same subject we recommend the books by Volkovyskii et al, Schwerdtfeger, and Flanigan. Review by Palle Jorgensen, August 5, 2006.

Great Book on complex analysis
Last year i took a graduate course on complex analysis having very little previous knowledge about it, cause i study physics and this was the book i used to help my self. This book resulted being a delight, it's wonderfull the way it is written the clarity, all theorems with proofs and starts from the basics till more advanced topics. I recently read it completely and the same thing happened: is a wonderfull book, my plan now is to get the bigger work of Markushevich, to extend the knowledge. In one word buy it you won't regret it!

Lacking in examples and reasonable exercises
We used this text for a one semester undergraduate course in complex analysis at North Dakota State, and the entire class HATED the text. There are few exercises and they are of poor quality. The text also contains few examples, and they are often completely missing when they are most needed. I don't even consider this text as having reference value in the future. Yes, it is a condensation of Silverman's translation of Markushevich's three volume work. One might be better off with the whole book, because this edition seems to be missing a lot. Our professor finally gave up on the text and spent twice the usual time preparing his lecture notes so that we wouldn't have to open the text ever again, and I don't plan to. ... Read more

Product Description
With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle.

With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory.

Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences.

The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory. ... Read more

Customer Reviews (9)

Nice choice of topics
If possible, I would give this book 4.5 starts, just because I don't think it is quite as good as the great classic by Conway, "Functions of One Complex Variable" (which treats all the standard topics).On the other hand, Stein and Shakarchi's book is beautiful, lucid, and obviously written by one of the grandmaster analysts of our time.Also, I give it five stars for including a beautiful treatment of the Paley-Wiener theorem, a topic that doesn't usually make its way into elementary complex analysis texts.Finally, this book has the best treatment I've seen of the Hadamard factorization theorem.

Good
Exactly same item as I expected.
This book is a little hard for undergrad student.
There is no solution for exercise question.

Informative, but too dense
The text packs a lot of information very tightly, making it difficult and slow to read. As a textbook or reference material, it works fine, but expect to do most of your learning in class rather than by reading the book.

it is just good
I got a copy of this book. It is a text for undergraduate students in pure mathematics.It is a good reference for elementary proofs of most common theorems in complex variables. However, some important theorems (ej: Three lines lemma and Picard theorem) are placed as exercises and problems. It is not a book for applications in engineering, its applications are taken from number theory.At some places it refers to sections or chapters in other books in the Princeton lectures in analysis. I think this is a four starts book.

The exercises are not very good
I used this book in a first year graduate course.I found the exposition not very clear, and the exercises particularly uninteresting.If you have the choice, I definitely recommend Gamelin's Complex Analysis instead. ... Read more

Product Description
This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. ... Read more

Customer Reviews (23)

Great Learning Experience
This is truly a well-crafted book. The organization is tight and the book is largely self-sufficient, really only calling upon material covered in his previous book, Principles of Mathematical Analysis. Rudin only proves such lemmas as he needs to get to the major results; peripheral facts and theorems are left to the exercises, which are far from routine and require a lot of thought and patience to get through. The book is rather terse, but there's a powerful elegance behind it. The proofs are rigorous and avoid hand-waving, albeit Rudin intentionally leaves gaps for the reader to fill. Working carefully through this book and doing as many exercises as possible, one can get a good grasp of the material.

However, this is not a book I would recommend for people who have not had prior experience with measure theory or complex analysis. Rudin provides little motivation. His arguments are elegant, but his methods are designed to give the quickest route to the results, which is seldom the most intuitive one. For example, where other authors explicitly construct Lebesgue measure, Rudin opts for a difficult proof of the Riesz Representation Theorem from which he pulls Lebesgue measure as a magician might pull a rabbit out of a hat. A lot of the material has a lot of geometric intuition behind it, but Rudin seldom goes out of his way to point it out. The beginner would be much better served getting an introduction to the material elsewhere before tackling Rudin. It will enable one to get much more out of the book than slogging through it blindly. It's one thing to know the definitions and be able to follow the proofs, and another thing to understand what it all means; Rudin doesn't hold the reader's hand where the latter is concerned.

For those with adequate preparation and sufficient daring, this book makes for a great learning experience and I daresay is even fun in places.

One of a Kind
I normally don't review books that already have this many reviews, especially when I agree so much with the reviews that already exist.But I'm teaching out of green Rudin for the first time this semester, 20 years after getting to know the book well as a student, and I find myself so enthusiastic about it again, that I just had to chime in with an "Amen" to the other positive reviews.When it comes to mathematical writing, it doesn't get any more exquisitely elegant than this.

Probably all our reviews are irrelevant, however, because there are probably very few discretionary purchases of this book:There will be nearly a one-to-one correspondence between buyers of the book and students in classes for which it is required.For them, I can only recommend skipping the outrageously expensive hardback (which even at its high price is pretty cheaply constructed nowadays) and opting for the more reasonable international paperback edition.

Rewarding, but has limitations
This is a very nice book, but in my opinion it is the worst of Rudin's three well-known books on analysis.The third, Functional Analysis, is a better representation of the subject, and (as opposed to this book) contains applications of analysis to other areas of mathematics, such as number theory and PDE.

The chapters here are very good for reference, but they are not well-written (at all) for self-study.So my two cents is: it's good, but frustrating at times. Rudin has a talent for making difficult things clear, but he has a more sinister talent for making simple things appear difficult.

I love this book!
I love this book, even though I have not absorbed more than a small portion of it yet.I find this to be a much better book than the "baby Rudin", which struck me as dry, overly concise, and without motivation.This book provides ample motivation, and although it proceeds in great generality, proceeds at a reasonable pace.

The best thing about this book, however, is the spirit of it--the integrated approach to analysis that Rudin takes is unique and greatly appreciated--Rudin is, like Lang, a testimony to the fact that the best mathematicians do not draw artificial lines between different areas within mathematics.Rudin presents the material in ways that connect to other areas of mathematics and will help the reader become a better mathematician, even if she never directly uses any of the material contained in this volume.

I would not recommend this book as a first exposure to measure theory or complex analysis--it is advanced and requires a great deal of background to fully understand and appreciate.But I think this is a book that any serious mathematician should add to their collection and eventually work through.People wanting to learn measure theory might look to the book by Inder K. Rana, or to the classic book by Royden.For more elementary treatments of complex analysis I would recommend the classic by Ahlfors, Theodore Gamelin's book, or the book by Greene and Krantz.

My 2 cents
There are some excellent reviews here for this outstanding book, so I will try to avoid repetition. In preparation for my qualifying exams in graduate school, two of my colleagues and I did all of the exercises in Rudin (give or take a couple, no more). What I found striking at the time was how Rudin took three subjects -- measure theory, functional analysis, and complex analysis -- and weaved them together seamlessly. It is not that I believed them to be separate subjects, but until then I hadn't realized just how they all fit together. Really, this book is superb.

A word of warning, though. Rudin's prose is concise, and his proofs leave you wondering if you'd ever be able to reproduce them on your own. It is what we in the business are used to call 'elegant'. It pays to work in groups, persevere, and go over everything twice or more. Good luck. ... Read more

Product Description

The guide that helps students study faster, learn better, and get top grades

More than 40 million students have trusted Schaum's to help them study faster, learn better, and get top grades. Now Schaum's is better than ever-with a new look, a new format with hundreds of practice problems, and completely updated information to conform to the latest developments in every field of study.

Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!

Schaum's Outlines-Problem Solved.

Customer Reviews (17)

Cannot get better
I am a huge fan of books by Schaum. This book is no exception - very advanced level examples, to the point, and yes - very inexpensive. Thanks, I passed by exams because of you.

excellent
i still remember one winter vacation decades ago, when i learned complex analysis all by myself using this book. i, a physic major then, learned everything i need to know (and more) in one month. i thought i'm a genius. but probably it's just because Spiegel is an excellent textbook writer ~~

School book
This book is going to be used to help me understand a book that I bought.Hopefully it will help.

An excellent supplement for the learning of complex variables
The first graduate level course in mathematics that I took was complex variables. Despite having been very successful as an undergraduate, I felt a bit of trepidation, as the rumor was that graduate school was much, much harder. To help ease the transition, I bought this book as a crutch, which was a wise move. I found problems similar to the assigned homework problems in here and by working through them; I was able to figure out how to do the homework. As a consequence, I was able to do nearly every homework problem, missing only a few points due to minor errors.
That work also led to my achieving very high scores on the exams and getting an A for the course. I strongly recommend this book to anyone who is taking complex variables and is having a bit of difficulty.

Just the Beef
I never took a course in complex variables so a professor recommended this book to me.The format is simple and straighforward.Every chapter begins with a terse exposition of the subject matter to be covered.Immediately following is a longer section of "solved problems," where the theory is put into use.The final section consists of problems for the student to solve.The techniques necessary to solve these problems are covered in the earlier "solved problems" section.It is apparently intended as a supplementary textbook, although to me the book seems perfectly adequate by itself for self teaching.No good if you like to learn through reading alone, but for the hands-on type it is very good. ... Read more

Product Description
The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing Ph.D. qualifying exams in complex analysis. ... Read more

Customer Reviews (12)

Amazing book
There were many typos and wrong answers at the back of the book which confused me sometimes, but besides this, I've never seen a more complete and readable book on complex analysis.He proves every theorem (although sometimes the proof comes several chapters after he uses the result) and provides plenty of examples for each new concept.Gamelin even has graduate-level complex analysis in this book too.

Great for engineers...not for mathematicians
This book was used for my undergraduate complex analysis course. I struggled with the author's lack of clarity and ended up getting a B.Then just last month I picked up a more formal treatment of the subject--Serge Lange's Complex Analysis--and felt like I learned more in one week than I did in a whole semester with Gamelin.

Short of providing formal proofs, the text does not even provide definitions.For example, the meaning of "homotopy", which can be made mathematically rigorous, is left totally to intuition.For a student accustomed to the definition-theorem-proof structure, this text can be extremely frustrating.

A fascinating but at times cryptic work
I like this book.The author's presentation style is very clear and very friendly.However there are times when I wish the illustrations could be a little more clear and a tad larger.Consider the image on page 67.Shouldn't this be labeled a bit better?

I must confess: What I'm looking for (truly) is a complex analysis coloring book.While this ain't that, it's worth the read and gives a different view of the subject than Churchill and Brown.

A worthy book for my collection even if my 64 Crayolas sit in my drawer, unused.

A good introduction for any level
There are plenty of Complex Analysis books to choose from, but I really like this one.The exercises are very interesting and there hints for most of the more complicated ones.I've used this book in both undergraduate and the first year graduate courses, and it's been pretty consistently enjoyed.

Not my taste
Although I can see what others might like in this book, I did not care for it. (To be fair, I am not sure how much of this is the book's fault and how much is the fault of the subject.) I was looking for something a bit more mathematical, and more along the lines of (say) Rudin's real analysis, and instead this book was less formal than I would have liked, seemed geared toward applications of the material in physics and engineering, and was more calculus-oriented. (Maybe the latter is inherent in the subject, I don't know.)

I prefer mathematics textbooks written in the definition-theorem-proof (followed by examples) style, and this book is not that. Terms are sometimes defined only intuitively (I don't mind intuition in addition to a formal definition, but I do mind when it is in place of a formal definition), and there are no "marked" proofs (instead, proofs are supposed to follow from the surrounding discussion, which is sometimes formal and sometimes less so).

The index was awful. I was looking for a proof of the fundamental theorem of algebra and found only one reference: to page 4, where the author promises that we will see many proofs of this theorem. (Where? Who knows!) ... Read more

Product Description
This is the fourth edition of Serge Lang's Complex Analysis. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a gradute course a great deal of flexibility in structuring a more advanced course. ... Read more

Customer Reviews (6)

Doesn't seem like a graduate book
I really have two big gripes with this book: (1) it is too slow and (2) there are not enough exercises and the ones that are given could have been better.

I bought this book to study for my quals, but found myself abandoning it after encountering tedious and unnecessary proofs and insufficient/trivial exercises.

It might, repeat *might* be good for beginners, but anyone whose taken advanced calculus (as just about any grad student would have) will be frustrated with the slow pace first few chapters.Frustration will continue after getting to the end of a complicated chapter and finding no excercises (!).

I also thought the orginization of the material was really unnatural (e.g., why are fractional linear transformations discussed so late in the book?).

I don't know... If you understand theory with out any problems to practice on, then you might be able to use this book.

A very good text
Lang's Complex Analysis is an very good text for anyone wanting to move beyond introductory complex analysis.

Excellent, but inconsistent pace, unnecessary proofs in early chapters...
There are about as many opinions on this book as there are different books that Lang wrote, but there is a reason for this: this is one strange book, even among Lang's.

I will start out by saying what I like about this book: most of it.This book provides a lot of topological flavour to complex variables, which I find very helpful.To someone who thinks topologically, many of the proofs in this book will seem more intuitive than in other texts.This is particularly true when you get into more advanced material.

Overall, the writing is very clear.Lang is excellent at providing motivation, especially as you get farther along in this book.Unlike some of his other books, he can't be criticized as moving too fast in this book.

Now the bad: the book starts out very slow, painfully so.It seems the first chunk of the book is aimed at teaching rigorous complex analysis to someone whose background in analysis is weak.Lang repeats all of the basic theorems about limits, differentiation, convergence, etc. in full detail.However, the material picks up eventually, and by the end of the book it's moving fast enough that anyone who enjoyed the first part will have trouble understanding the later material.This book covers a lot more material than most undergrad books on the subject, so I suppose it lives up to the GTM title.

Bottom line: I don't like the choice or order of topics in initial chapters.Some of the "new" material specific to complex variables is mixed in with old results common to basic analysis on the real line.Anyone with a good background in analysis will be frustrated trying to find what they need to learn.Also, Lang confuses the logic of the subject by working with the terms "analytic" and "holomorphic" separately for a great deal of time before showing their equivalence.His definitions, terminology, and development don't line up with many other authors, and he has not convinced me that his choice of development was justified...because most of the stuff I like in this book comes after the first few chapters.However, if you can get past these hurdles, you'll find that this is a pretty great book that has a lot to offer.

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. ... Read more

Product Description
This book provides a comprehensive introduction to complex variable theory and its applications to current engineering problems and is designed to make the fundamentals of the subject more easily accessible to readers who have little inclination to wade through the rigors of the axiomatic approach. Modeled after standard calculus books--both in level of exposition and layout--it incorporates physical applications throughout, so that the mathematical methodology appears less sterile to engineers. It makes frequent use of analogies from elementary calculus or algebra to introduce complex concepts, includes fully worked examples, and provides a dual heuristic/analytic discussion of all topics. A downloadable MATLAB toolbox--a state-of-the-art computer aid--is available.Complex Numbers.Analytic Functions. Elementary Functions. Complex Integration. Series Representations for Analytic Functions. Residue Theory. Conformal Mapping. The Transforms of Applied Mathematics. MATLAB ToolBox for Visualization of Conformal Maps. Numerical Construction of Conformal Maps. Table of Conformal Mappings. Features coverage of Julia Sets; modern exposition of the use of complex numbers in linear analysis (e.g., AC circuits, kinematics, signal processing); applications of complex algebra in celestial mechanics and gear kinematics; and an introduction to Cauchy integrals and the Sokhotskyi-Plemeij formulas.For mathematicians and engineers interested in Complex Analysis and Mathematical Physics. ... Read more

Customer Reviews (6)

Excellent Book
There are many books on complex variables, but this surely rates well as an introduction. It is great for self study. It bridges the gap nicely from calculus. The problems at the end of the sections are of a rich and varied type and do enhance your learning experience. This book deserves a second look.

A reference for life!
Complex Analysis is always there in every applied math document of engineering context. The reason I bought the particular book was that I stumbled on some old forgotten Conformal Mapping techniques in Digital Filter Design and needed some good reference to go through...I ended up reading the whole book from first to last page as it managed to capture my interest and distract me from my original purpose for a couple of happy months. So if you are planning to stick to the foundations beyond your studies and course exams, then THIS BOOK IS FOR LIFE...the subject is very extensive and tricky but the book manages to present completely all the necessary elements in the right pace and volume that keeps the application-oriented reader's attention focused while keeping at the same time -in my opinion- the right level of mathematical strictness. All the most essential theorems and formulas are nicely placed intro frames so underlining is not that necessary. Last but not least there is a wealth of examples and illustrations that make it a very friendly tool for anyone about to take course exams or some old engineering graduate seeking a quick reference like myself.

Good reference
This book was not exactly introductory level but if you have some familiarity with concepts, it will serve as a good reference book. Very concise but contains many good examples.

I used this book in conjunction with "A First Course in Complex Analysis"
by Dennis Zill for a graduate level course, which is more of an introductory text than this book.

I recommend using both for your first course.

Another reference: Search for "Complex Analysis Modules by Mathews") on google. This served as a great online reference and has a corresponding book: COMPLEX ANALYSIS: for Mathematics and Engineering, Fifth Edition, 2006 by John H. Mathews and Russell W. Howell. Although I did not read this book, the author has put up wonderful online notes from this book, which I did use.

Excellent Book!
First let me say that this book was an introduction to the subject for me. After reading the first six chapters, and working through most of the problems, I have to say this book is great. I highly recommend this to anyone who is learning on there own. In particular, the chapter on residues is excellent. The chapter on series is also good, although I would have liked more worked examples for proofs involving uniform convergence. Also, a little more emphasis on the Arguement would have been nice. Nevertheless, 5/5 for this one, it is extremely well written and the authors really provide motivation for the theorems to come. This is definitely one of the best math books I have read. Great buy, worth every penny.

Good Introductory Book
This was the book that I learned Complex Analysis from. Definitely made the subject accessible to pretty much any reader. Plenty of exercises: some more theoretical, some more applied. It skillfully straddles the gap between being a theoretical math book and a math book for people with more applied aims (such as engineers). Most topics are covered thoroughly, though certain more complicated subjects such as winding number are left out for simplicity.

This book definitely prepared me for tackling the dense, theoretical, and exceptional "Complex Analysis" by Ahlfors. I'd recommend it as an introductory book for anyone trying to get into the subject who is intimidated by Ahlfors, as well as for anyone who is only interested in the essential commonly-applied tools. ... Read more

Product Description
This volume on complex analysis offers an exposition of thetheory of complex analysis via a comprehensive set of examples andexercises. The book is self-contained and the exposition of newnotions and methods is introduced step by step. A minimal amount ofexpository theory is included at the beginning of each section in thePreliminaries, with maximum effort placed on well-selected examplesand exercises capturing the essence of the material. The examplescontain complete solutions and serve as a model for solving similarproblems given in the exercises. The readers are left to find thesolution in the exercises; the answers, and occasionally, some hints,are given. Special sections contain so-called Composite Examples whichconsist of combinations of different types of examples explaining someproblems completely and giving the reader an opportunity to check allhis previously accepted knowledge.
Audience: This volume is intended for undergraduate and graduatestudents in mathematics, physics, technology and economics interestedin complex analysis. ... Read more

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Customer Reviews (19)

Good, but could be better
This book has excellent examples and most of the proofs are very easy to understand.It's a great book to use if you're learning complex analysis on your own.

It's not, however, the only book you should use.I prefer math books that are more condensed with short, easy-to-read theorems.I would not list this book under such a category.In fact, many of it's theorems are so long, they require you to turn back and read other theorems pages earlier just to figure out what the necessary conditions are to use the theorem.I found this to be confusing and it slowed me down quite a bit.Therefore, if you have the same preference, I would suggest using this book only for its examples and exercises.

Good Intro to Complex Analysis
One problem some students had with the book was that it can get a bit too wordy with its explanations, making theorems/concepts more difficult to understand. Other than that, it is a great book for undergraduates as the exercises consist of not only proofs, but also numerous computational problems which students can work through to practice applying the theorems. Examples at the end of each section are great and helpful.

Good book for undergraduate study.
It's a good book. The wording is precise and clear, there are plenty of examples and diagrams, and it seems to be well organized.

Definitely of reasonable caliber for undergraduate study.

Perfect condition, shipped on time
I needed this book for a class.I got it new for half of my local bookstore's price.Quick shipping, perfect condition!

Hated it.
Used this book as an undergraduate... hated it... I kept using a little thin old edition of "Complex Variables and Applications" by Churchhill to actually teach math using english....Ironically the instructor who was teaching out of his notes followed churchhills presentation closer then this text.

The treatment of this subject in this text is just so horrid for a FIRST LOOK AT COMPLEX THEORY.No elegance to it what so ever... ... Read more

Product Description

Filling a gap in literature, this self-contained book presents theoretical and application-oriented results that allow for a structural exploration of complex networks. The work focuses not only on classical graph-theoretic methods, but also demonstrates the usefulness of structural graph theory as a tool for solving interdisciplinary problems. Applications to biology, chemistry, linguistics, and data analysis are emphasized.

The book is suitable for a broad, interdisciplinary readership of researchers, practitioners, and graduate students in discrete mathematics, statistics, computer science, machine learning, artificial intelligence, computational and systems biology, cognitive science, computational linguistics, and mathematical chemistry. It may also be used as a supplementary textbook in graduate-level seminars on structural graph analysis, complex networks, or network-based machine learning methods.

Product Description
Revised and updated, the new Fifth Edition of Complex Analysis for Mathematics and Engineering presents a comprehensive, student-friendly introduction to Complex Analysis.It's clear, concise writing style and numerous applications make the foundations of the subject matter easily accessible for students and proofs are presented at an elementary level that is understandable by students with a sophomore calculus background.Believing that mathematicians, engineers, and scientists should be exposed to a careful presentation of mathematics, attention to topics such as ensuring required assumptions are met before the use of a theorem or algebraic operations are applied.A new chapter on Z-Transforms and Applications provides students with a current look at Digital Filter Design and Signal Processing. ... Read more

Customer Reviews (8)

Helpful Book
As a self-learner, I was looking for a self-contained book that would teach the fundamentals of complex analysis in a straight-forward manner without getting bogged down with a lot of advanced level mathematical jargon and proofs on every page.I was also looking for a book that would show how complex analysis is applied to various physical problems.I found all of this nicely packaged in this book.The explanations for the various topics were clear, with a lot of geometrical interpretations so that you could picture what was being discussed.I found the coverage of Riemann surfaces to be a good example of this approach.There are plenty of worked examples to help deepen your understanding of the material.The problems at the end of each chapter were not extremely difficult, yet helped to reinforce or extend the chapter's concepts.For a self-learner, this is a helpful approach.It not only builds understanding, but self-confidence.I found out about this book after trying to learn complex analysis with McMahon's book Complex Variables Demystified. I lost confidence in the ability of this book to provide clear explanations of the topics after discovering all of the serious errors contained within.I had to turn to the internet for clearer explanations, and found the website complement to this book.I was impressed with the website, so bought the book.I was not disappointed.

Complex Analysis for Mathematics and Engineering
This book was in great condition and the company worked well with me in the return of the book since it wasn't exactly what I needed.I would definately use them again.

Get your feet slightly damp in analysis
This book is a great way to ease into some more advanced topics in mathematics.For instance, within the first chapter you start to study topology of the complex field (mappings of sets of points, bounded sets, domains, etc).Of course none of this can be done until you establish the complex number system in terms of R which we are familiar with, and defining the arithmetic options and such.The construction of the complex field in terms of R and continually building off this idea the entire book was very fun and intuitive.It is a great help to read the semester before you take a more rigorous course in analysis, because it will give you a peak at its subtleties without beating you over the head with them.

An understandable presentation of complex analysis.
This book is useful for learning both the theorems and applications of complex functions.It is better organized and more up to date than other books I found in the library.The proofs and examples are complete and easy to understand and there are many well composed figures which help illustrate the concepts. I learned a lot about complex sequences and series and enjoyed the many practical examples like conformal mappings. The computer supplements for Maple and Mathematica look interesting. I can see how complex analysis is used in the real world.

Should be renamed Complex Analysis for Engineers
If you are engineering student, then you're going to love this book to death. If you are interested in pure mathematics, on the otherhand, then do not waste your time here. This book promotes absolutely no rigor whatsoever. I'm using this textbook for my complex analysis course right now, and I must say that this book is beyond boring. I have to force myself to read the book mainly to keep up with my class. I feel that doing rigorous mathematics gives the reader a sense of freedom and that they are free to do things they never thought was possible in the mathematical world. This book, on the otherhand, forces the reader to pretend like they are doing calculus I work. Been there, done that, time to move on.

Bottomline: Avoid this book if you love pure mathematics(if you have to use it for a complex analysis course, then pick up a supplmentary theoretical text that lets you enjoy the subject). ... Read more

Product Description
The new Second Edition of A First Course in Complex Analysis with Applications is a truly accessible introduction to the fundamental principles and applications of complex analysis. Designed for the undergraduate student with a calculus background but no prior experience with complex variables, this text discusses theory of the most relevant mathematical topics in a student-friendly manor. With Zill's clear and straightforward writing style, concepts are introduced through numerous examples and clear illustrations. Students are guided and supported through numerous proofs providing them with a higher level of mathematical insight and maturity. Each chapter contains a separate section on the applications of complex variables, providing students with the opportunity to develop a practical and clear understanding of complex analysis. ... Read more

Customer Reviews (5)

Physics major's perfect companion
This book is the best book for self study I found. Now that it has a self study guide with every other odd problem worked it even adds to its value. I took this course years ago and forgot most of it. I needed contour integrals for quantum field theory. This book also includes complex fourier analysis. Starts off with polar forms of complex numbers. Gets into a little complex analysis, but not too deep to not be able to follow. This book is the perfect book for anyone taking quantum mechanics or quantum field theory. Includes hermitian matrices. It then goes into complex differentiation and integrals. This book is about the level of looking back at a calculus II book after taking differential equations. The authors explain things at a level of a good calculus book.

Best Book on Complex Analysis
( You can see a sample of this book on Google Books (just type in the books name), also a sample of the Student Guide).I am taking this course in fall 09 and I can say that having been assigned many bad books throughout my courses you really appreciate when you find one that is as perfectly written and presented as this one.The book my instructor makes us get is the worst thing I have ever seen.It does not explain anything just presents definitions, theorems, and proofs then says here is 2 examples (which have nothing to do with what you are trying to learn) then gives you 10 problems and says do them (but isn't clear on its instructions). My instructor presents in this same manner so I can see why this was the book chosen.A few people in the class are getting outside help but won't say from where so this book is a lifesaver.Unfortunately I found it too late, but I hope you will find it in time if you take this class.I strongly recommend this book over all others for anyone (not just in classes) who wishes to learn complex.

Great introductory text on Complex Analysis
I used this book for an undergrad course in Complex Analysis.This book is great for undergraduates looking for some sort of exposure to the realm of complex analysis and is is especially great for students of Physics and EngineeringThe book covers the basics of analytic function theory, Cauchy-Riemann Equations, Cauchy Integral Theorems, Series and Sequences, Conformal Mapping, Residue Calculus, as well as a section of applications at the end of every chapter.The book is very readable, and a lot more beginner-friendly as opposed to Alfhors, which would be the next great read for the student interested in complex analysis at the graduate level.Overall this is a great intro to some really beautiful mathematics.I know I enjoyed the book.

Great book for engineers
I used this book in conjunction with "Fundamentals of Complex Analysis..." by Saff et al for a graduate level course.

This book gives very clear introductions and explanations of complex variable concepts and served as a boon for my first complex variable course; I went through many other books but they all seemed to be much more abstract that this one. If you're new to the world of complex variables and have trouble reading existing books, this book may very well be your life saver.

Another reference: Search for "Complex Analysis Modules by Mathews") on google. This served as a great online reference and has a corresponding book: COMPLEX ANALYSIS: for Mathematics and Engineering by John H. Mathews and Russell W. Howell. Although I did not read this book, the author has put up wonderful online notes which I did use.

Excellent supplment to Brown and Churchill
I used this book for two Graduate semesters of Complex Analysis. The course text was Brown and Churchill which I often found lacking in detail. This book might not be consider by some as a Graduate level text however I found it to be an excellent supplemental text to fill in the gaps and improve my understanding of the material. ... Read more

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Customer Reviews (4)

plane geometry and complex numbers
I discovered this book some twenty years ago while trying to improve my knowledge of plane geometry; I used it especially to work on circle pencils: a part of geometry I had already encountered time and again; setting up circles through two-rowed hermitian matrices and linear transforms {z->(az+b)/(cz+d) }as done in the book is both very pretty and efficient. The appendix (numbered 3) describing the use and applications of the characteristic parallelogram really appealed to me. I was also quite impressed by the way the cross ratio of 4 complex numbers is dealt with in the book; to put icing on the cake, one can find within those 200 pages some knowledge of non euclidian plane geometry plan...and dynamical systems associated with linear transforms in the complex plane; very informative and quite refreshing.

a good beginning
Schwerdtfeger's nice little book starts at the beginning with geometry of circles, Moebius transformations (a third of the book), and it covers some selected aspects of complex function theory, but the emphasis is on elementary geometry.Harmonic and analytic functions are only touched peripherically.

The central topics are (in this order): geometry of circles, Moebius transformations, geometry of the plane, complex numbers, transformation groups, a little hyperbolic geometry, and ending with a brief chapter on spherical and elliptic geometry.

The book was published first in 1962, but reprinted since by Dover. It is suitable as a supplement in a standard course in complex function theory, at the late undergraduate level, or perhaps at beginning graduate. While it contains attractive geometric concepts, it leaves out a systematic treatment of power series. Some readers might want to begin with that; using some of the other Dover titles on complex functions. We recommend the books by Volkovyskii et al, Flanigan, and Silverman. Review by Palle Jorgensen, August 5, 2006.

This book contained the stuff I wanted to know
I was interested in projecting a network onto hyperbolic space using the upper half plane projection. This book contained the equations relating to that, particularly the moebius transformation z' = (az+b) / (cz + d), and also stuff on stereographic mapping which I found useful.

I have not taken the trouble to understand much of the more in-depth parts of the book, but it is so clear and step-by-step that even though I am not a math student, I'm fairly confident that I could. The whole thing was fairly mind-opening.

Interestingly, after reading this and developing my own intuitions (eg: that flat translation, rotation and scaling are special cases of parabolic, elliptical and hyperbolic transformations with a fixed point at infinity), a re-reading discovered these conclusions in the book. So you can take the exposition and run with it. What I'd really like is to be able to get the n'th root of a transformation (to animate them). I suspect that that's in there too.

The book does not cover real-world applications (aerodynamics, electrodynamics), but that's cool. It's purely about the math.

Should be a "must read" for math students
This inexpensive book covers material not easily found elsewhere but key in understanding complex functions. The problem with complex functions is they are hard to visualize because the input is a plane and the output is another plane. The book covers Circles, Moebius transforms, and Non-Euclidean Geometry. The level is senior undergraduate, 1st year graduate. The book is easy to understand with good exercises. I really like this book. ... Read more

Product Description
A complete guide to carrying out complex survey analysis using R

As survey analysis continues to serve as a core component of sociological research, researchers are increasingly relying upon data gathered from complex surveys to carry out traditional analyses. Complex Surveys is a practical guide to the analysis of this kind of data using R, the freely available and downloadable statistical programming language. As creator of the specific survey package for R, the author provides the ultimate presentation of how to successfully use the software for analyzing data from complex surveys while also utilizing the most current data from health and social sciences studies to demonstrate the application of survey research methods in these fields.

The book begins with coverage of basic tools and topics within survey analysis such as simple and stratified sampling, cluster sampling, linear regression, and categorical data regression. Subsequent chapters delve into more technical aspects of complex survey analysis, including post-stratification, two-phase sampling, missing data, and causal inference. Throughout the book, an emphasis is placed on graphics, regression modeling, and two-phase designs. In addition, the author supplies a unique discussion of epidemiological two-phase designs as well as probability-weighting for causal inference. All of the book's examples and figures are generated using R, and a related Web site provides the R code that allows readers to reproduce the presented content. Each chapter concludes with exercises that vary in level of complexity, and detailed appendices outline additional mathematical and computational descriptions to assist readers with comparing results from various software systems.

Complex Surveys is an excellent book for courses on sampling and complex surveys at the upper-undergraduate and graduate levels. It is also a practical reference guide for applied statisticians and practitioners in the social and health sciences who use statistics in their everyday work. ... Read more

Customer Reviews (1)

Perfect fit to my best hopes for this book
Dr Lumley has been making the R package, "survey",available for several years. It adds functionality that had previously only been available with expensive commercial statistical packages This book contains a wealth of useful examples, both of functions within the survey package and of their use with relational database interface packages such as RODBC and DBI. I was particularly pleased to find the NHANES datasets well-represented in the many worked examples throughout the book chapters and Appendices. Lumley had announced its forthcoming availability last Fall on the r-help mailing list and it proved to be exactly what I need to improve the validity of my work with those publicly accessible datasets.

--
David Winsemius, MD, MPH ... Read more

Product Description
Modern Real and Complex Analysis

Thorough, well-written, and encyclopedic in its coverage, this text offers a lucid presentation of all the topics essential to graduate study in analysis. While maintaining the strictest standards of rigor, Professor Gelbaum's approach is designed to appeal to intuition whenever possible. Modern Real and Complex Analysis provides up-to-date treatment of such subjects as the Daniell integration, differentiation, functional analysis and Banach algebras, conformal mapping and Bergman's kernels, defective functions, Riemann surfaces and uniformization, and the role of convexity in analysis. The text supplies an abundance of exercises and illustrative examples to reinforce learning, and extensive notes and remarks to help clarify important points. ... Read more

Product Description
This unusual and lively textbook offers a clear and intuitive approach to the classical and beautiful theory of complex variables. With very little dependence on advanced concepts from several-variable calculus and topology, the text focuses on the authentic complex-variable ideas and techniques. Accessible to students at their early stages of mathematical study, this full first year course in complex analysis offers new and interesting motivations for classical results and introduces related topics stressing motivation and technique. Numerous illustrations, examples, and now 300 exercises, enrich the text. Students who master this textbook will emerge with an excellent grounding in complex analysis, and a solid understanding of its wide applicability. ... Read more

Customer Reviews (8)

Best Undergraduate Intro to Complex Analysis
Frankly, when I first learned complex analysis, I found it to be highly confusing, with the results bizzare and not intuitive even after long arduous proofs.This is probably the book that unlocked that confusion for me.Its slow start from analytic polynomials quickly moved into analytic functions.The link between path integrals and differentiation is nicely done through the means of first proving the rectangular version of the Cauchy integral.All the main results of introductory complex analysis falls naturally from that.Interestingly, the second half of the book then touches upon interesting topics in a very delightful way: conformal mappings, sample series and integrals, the Laplace equations and the Drichilet problems, and lastly non-power series and the Zeta functions.

Excellent Introduction to Complex Analysis
If you want to learn Complex Analysis, start with Schaum's Outlines. Then when you want to learn the methods and thinking of complex analysis, read this book. It's concise and gets the MAIN POINTS across in a friendly way.

Except for the topological stuff (they simplify things to avoid lengthy tedious discussion) this book is EXCELLENT. I disagree with the reviewers who said this book deals with things in too elementary a way. In fact it gives more general results and the REAL reasons behind complex numbers. Most importantly, it gives you a CONSISTENT FEEL for complex analysis techniques and concepts! For example, whereas most books treat a special case of the Riemann Principle of Removable Singularities where f is bounded. They use slightly tedious estimates of the Laurent coefficients to show that the terms with negative indices are all zero. This book simply shows that if lim (z-w)f(z) = 0 as z->w, then f(z) has a removable singularity, by appealing to the Schwartz Reflection Principle it proved earlier in the book. A more general result and gives a more integrated feeling for the theory.

ALL IN ALL A GREAT RESOURCE. After this, you can read Alfohrs, and then spcialized books on whatever. Lang is okay too but his results are not as general or intuitive as this book, and he uses power series constantly and is good for people who want a different perspective.

Good book
This book is excelent for a basic Complex Analysis course. It is very well writen, and the examples help you to understand the theorems. The book doesnt have to much solved examples, sometimes you need them.
I recommned to the complex Analysis book writen by Palka.

Hard to follow, not comprehensive enough
This book is disappointing, especially after encountering Newman's "Analytic Number Theory", which is a wonderful book.This book takes the readers on a concise, linear journey through Complex analysis to a few key theorems at the end, but does not do justice to the richness or diversity of the subject.This book will be especially lacking to students studying complex analysis for purposes related to applied mathematics.

The prose in the book is clear, but at times, as early as chapters 2 and 3, the equations are dense for an undergraduate text, with some steps less than obvious.There is a lack of motivation for the direction of development chosen in chapters 4-6, possibly a little of 2, 7, and 8 as well.Results are proven three or more times in cases of increasing generality.While this makes the theorems easy to follow, the redundancy may be confusing for a student studying the material for the first time.The authors do not provide much of a preview of what is to come, as I think authors of an undergraduate text should (and many, such as Gamelin, do).

This book is so small and compact that I question the authors' judgment in leaving out these various explanations--little would be lost and much gained by additional explanations.This makes me wonder what the intended audience is.I think anyone who is able to follow this book without trouble would also have no trouble following a more advanced and comprehensive book.There are a number of more advanced books that are actually much easier to follow.This brings me to my next comment:

This book leaves out a lot of important topics; it is far from comprehensive.There are not very many exercises either, and the exercises are mostly related to the material in simple ways.

For those studying complex analysis for the first time, I would recommend the Gamelin book over this one; its proofs are much easier to follow, it contains much more explanatory prose.It moves slower but it is much more comprehensive and covers more advanced material, and it is better suited to students with diverse interests and different backgrounds.I also recommend the Churchhill text as a straightforward book covering the basics.Advanced students might want to use the classic Ahlfors text.

perhaps the best introduction to complex analysis
This is the book that really made me understand basic complex analysis.It doesn't try to give the most sophisticated or slickest presentation for experts.Instead, it gives a beautiful, concrete, down to earth explanations.The best feature is the applications.D. J. Newman is one of the world's great problem solvers, and this book includes numerous examples of how to use complex analysis to solve problems in surprising ways.Even in the more standard applications, such as summing series, the book gives many unusual examples.It concludes with Newman's proof of the prime number theorem, which is substantially shorter and clearer than many other proofs. ... Read more

Product Description
Ideal for a first course in complex analysis, this book can be used either as a classroom text or for independent study. Written at a level accessible to advanced undergraduates and beginning graduate students, the book is suitable for readers acquainted with advanced calculus or introductory real analysis. The treatment goes beyond the standard material of power series, Cauchy's theorem, residues, conformal mapping, and harmonic functions by including accessible discussions of intriguing topics that are uncommon in a book at this level. The flexibility afforded by the supplementary topics and applications makes the book adaptable either to a short, one-term course or to a comprehensive, full-year course. Detailed solutions of the exercises both serve as models for students and facilitate independent study. Supplementary exercises, not solved in the book, provide an additional teaching tool. This second edition has been painstakingly revised by the author's son, himself an award-winning mathematical expositor. ... Read more

Customer Reviews (3)

Superb book for the mathematically mature
This is a master class in complex analysis. It goes way beyond most comparable books, but you have to be mathematically mature to get this (you should have met real analysis, and dated it). Just to take one example, the presentation of the proof of Cauchy's theorem is superb; trotting alongside the master, you see what is really going on. The author's casual asides, sprinkled throughout the text, are stimulating. The footnotes are a treasurehouse for those with some curiosity about the history of the subject. Of course, this is not an efficient Schaum-series style textbook; that explains the orthogonal ratings given to this book by the two preceding reviewers. If you want to get good grades at some exam, there are plenty of other books on the market that will do the job. But if you have dispensed with such mundane concerns and want to really understand complex analysis, buy this book as I have.

Examples are poor and low readability
The book was used at UCLA recently and I found it not enjoyable to read at all.The examples were very poor and did not seem to relate to the topicscovered.Not a book I would recommend.

An adequate text in complex analysis
This was the text used by our class at the State Universityof New York at Buffalo for complex analysis.Most of usfound the text to be too highly condensed.In other words, we were dissatisfied with the time spent on explanations, especially for some of the subtleties that can arise. A fine selection for a reference text, but, somewhat unpleasant as a text for teaching oneself the subject. Admittedly, complex analysis is a difficult topic.The author (a famous mathematician) does very well, actually, in the space alloted.(text by Ahlfors is superior, though) ... Read more

Product Description
This volume contains all the exercises, and their solutions, for Serge Lang's fourth edition of "Complex Analysis," ISBN0-387-98592-1. The problems in the first 8 chapters are suitable for an introductory course at the undergraduate level and cover the following topics: power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings, and harmonic functions. The material in Chapters 9-16 is more advanced. The reader will find problems on Schwartz reflection, analytic continuation, Jensen's formula, the Phragmen-Lindeloef theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and Zeta function. This volume also serves as an independent source of problems with detailed answers beneficial for anyone interested in learning complex analysis. ... Read more

Customer Reviews (3)

I have little taste for answer books, especially when they are wrong
I gave a take home test of problems from Lang, because the answers are not in the back of the book.A student challenged my answer to one question, because Shakarchi says the answer is something different.I did not realize anyone had gone to the absurd trouble of publishing answers to the excellent exercises in Lang's book until then.But Shakarchi is wrong, having made a careless error.Reading books of answers to problems is like reading a book about physical exercise only worse, since not only do you not benefit from the exercise of doing it yourself, but you lose most of the potential benefit of the exercise after seeing the answer.At least with physical exercise you do not lose anything from watching it done if afterwards you do it yourself.Look on page 85 of this book where Shakarchi asserts that problems VI.1.26b and VI.1.26e have the same answer because the denominators have the same residues,Well how about the numerators?In 26e the z in the numerator implies that the terms of the laurent series of the quotient are not all of even degree anymore, so I claim (check it!I could be wrong too)the correct answer is that all residues in that case are equal to 2, not zero.(This comes from 2= 1/(1/2) the quotient of the coefficient 1 of the z in the top, and the first non zero coefficient 1/2 ofthe taylor series of 1-sin at /2.)So without having read any more of this book, even if this is the only error,I say it is almost worthless to someone really wanting to learn the subject.But if you are like someone who cannot exercise without watching a Jane Fonda video, and you do buy this book, please work the problems yourself, and do not just take the author's word for them.I do recommend Lang's book, and the exercises. Probably my favorite complex book is that by Henri Cartan available in a cheap paperback, and the one by Frederick Greenleaf (available used and in libraries) is excellent for beginners, and he gives the answers (but not the solutions) to most of his exercises right in the book if you want that.

INCOMPLETE EXPLANATIONS
This book cannot be used without purchasing the actual book which it represents. This book has some solutions for another complex analysis book. What I thought was, that this book is similar to something like Schaum's solved problems (which is independent and not dependent on another book). In many cases, there is no work shown at all, and simply the answers. So its really of no use and has a deceptive title. I already had 4 other books in complex analysis, so there was no need for me to purchase another book which it represents.

The Right Stuff !!!!!!!!!!!!!!
I bought this book with the goal to enhance my problem solving skills on complex analysis. I am a theoritical physics student and needed to gain a very comprehensive and operational understanding of the topic. This book had it all: it's very well written, very rigourous and gives an extra dimension to just reading a pure text explaining the theory of the subject. I wish we had this type of books in physics ! Lang's book and Shakarchi's solutions are the winning combination to learn complex analysis!!!!! 2 thumbs up ... Read more