Book Description
Now available in paperback, this successful radical approach to complex analysis replaces the standard calculational arguments with new geometric ones. With several hundred diagrams, and far fewer prerequisites than usual, this is the first visual intuitive introduction to complex analysis. Although designed for use by undergraduates in mathematics and science, the novelty of the approach will also interest professional mathematicians. ... Read more

Customer Reviews (28)

Feynman Unleashed!!!
Like an a Jack Nickelson impersonator, the impersonator can sometimes outdo, maybe even overdo, Jack.

Tristan Needham is impersonating Richard Feynman here.In fact, he does Feynman better than Feynman!Feyman was known for creating graphical represenations from various branches of calculus in a time when such diagrams had become taboo even in physics!Blame it on the pompous retards, once again they had nearly ruined something for everyone else by trying to imposing arbitrary and ultimately false order upon various physical situations. Their objective this time? It was visualization.

Before reading this book, I would definitely have my head firmly grounded in the fundamentals of applied math.I recommend 4 months or so of serious self-study in What is Mathematics by Courant andComplex Variables by Francis J Flanigan.

Innovative Text
This text is innovative to say the least.It has been very helpful to me in my work as an engineer and graduate study in engineering by providing a great deal of insight into difficult topics.One should note that this text includes topics more theoretical than those included in texts such as Brown and Churchill.

Beautiful
This is one of the best math books ever written. It is insightful, reader friendly,and has an excellent set of exercises. It also covers a broad range of topics including applications to physics.

Great Intuitive Coverage of Complex Variables
This is a great book for a Complex analysis course, especially when combined with a more traditional text like Ahlfors or Knopp's Theory of Functions.Needham's geometric approach builds 'feel' for the subject and manages to include a huge range of topics.Sometimes just trying to work out what you are seeing in a particular visualization can lead to new insights.All in all its an innovative and well written book.

Pretty but not a substitute for traditional text.
My slightly harsh rating is an antidote to all the gushing about
this book. It is a nice book with lots of pretty pictures and
genuine geometrical insights and is well worth reading as a
supplement to traditional complex analysis texts. The geometrical
topics are actually quite good. If you are a maths major then
this book will be of limited use because its coverage of the
traditional topics is simply too weak. The geometrical approach
quickly runs out of steam, in my opinion, once it gets into
complex integration. Homotopy does not even rate a mention in the
index. My pet dislike was the almost complete omission of the
calculus of residues. The author dimisses that topic as being
old-fashioned. True, the application to computing real integrals
is reduced since the advent of computers. But I think that a
maths major would need to be aware of Jordan's Lemma and other
techniques to estimate the asymptotic behaviour of integrals
along curves. I also found that the treatment of multi-valued
functions and branch cuts quite confusing, which is surprising
in a book which is supposed to have a strong geometrical focus. ... Read more

Book Description

Customer Reviews (8)

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.

A wonderful text -- Highly recommended!
I purchased this book as a reference book for my first analysis course.It is very well written, and easy to follow.Dr. Shilov has a very nice way of organizing this text:He puts all the definitions at the beginning of the chapter and the subsequent sections are results of those definitions.It makes for a very quick reference.His presentation of the included proofs is also very nice.There were several occasions I found myself thumbing through it for a second perspecitve.

As far as the actual material presented, Dr. Shilov starts off with funtions of one real variable, then rather quickly generalizes to complex variables and N dimensional functions, so you'll quickly see metric theory and some topology.He does keep in mind this is intended for undergrads and first year grads though.

Oh, another nice feature is the price!I'd recommend this book to any math enthusiast as a reference, or to someone going through an early analysis course. ... Read more

Customer Reviews (15)

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...

elegant treatment
The book reveals complex analysis as a very elegant and lovely branch of mathematics. The level of rigour is not that of Marsden's other book, Elementary Classical Analysis. Instead, Basic Complex Analysis can be usefully read by non-maths majors, especially those in physics and engineering.

Key ideas are well covered. Starting with the Laurant series, which generalises the Taylor series. Then, from this, the idea of contour integration is examined. Giving rise to the Residue Theorem and the winding number. All because the only term that does not integrate to 0 is 1/z, which gives the complex log and its imaginary argument is the only thing left. So simple and powerful. Amazing that an essentially arbitrarily intricate contour integral can be given by the residues at the enclosed poles! Yet the text's derivation should get straightforward to follow for most readers.

If you are going onto advanced physics, like quantum electrodynamics, then this theorem is used extensively.

The book also covers important subsequent ideas. Especially conformal mapping and the Schwartz-Christoffel transformation. The treatment of conformal mapping, though, is only a hint of the richness of analysis available here.

The numerous problems are also good for the student to tackle.

Very good book, actually
When I first started with this book, I was not a fan. However, the book grew on me over time. Marsden and Hoffman do a very good job of blending both theoretical and computational aspects of complex analysis. They do a very good job of motivating and explaining the proofs, and they do not leave out any details (this is both good and bad - it can distracting, but as long as you pay attention, you will never get lost). The illustrations in the text are for the most part illuminating and useful, and the worked examples at the end of each section are not bad as well.

I did have a few minor problems, though. While many of the exercises are good, some of them seemed rather trivial. The chapter on conformal mapping could use some work. The binding on mine started to come apart by the end of the semester, although that may have been my fault.

Quite Dry
This is the second book that I have read beside the Vector Calculus by Marsden and Hoffman.This book rushes you through with an introductory chapter and go right into the heart of complex analysis.The author assumes you to have a great professors that can explain things in detail when you can't quite understand what is written in the text.Unfortunately I did not have a great instructor.

The examples of the book are quite simple, compare to some end of section problems.

Overall this book has no surprises as it is quite dry, got bored from reading it.If it was not a required text book for a 3rd year complex analysis course, i wouldn't recommend it to anyone.There are many other books out there that are better written.

A versatile introduction to the subject.
I used an earlier edition of this text as an instructor 20 years ago.The students in my class at the time were equally divided among the fields of mathematics, physics, and engineering.The book proved to be quite useful for all of them.Marsden skillfully strikes a balance between the needs of math majors preparing for graduate study and the needs of physics and engineering students seeking applications of complex analysis.

The book is clearly written and well-organized, with plenty of examples and exercises.My only significant criticism of the first edition was the author's tendency to label many examples of contour integration as theorems.Technically, there is nothing wrong this, but I found that some of my students tended to memorize the statements of these "theorems" rather than focus on the methods of integration discussed (for example, "Pac-Man" integrals with branch cuts along rays other than the positive real axis).Nonetheless, this is a fine text that has--not surprisingly--continued to be widely used for over two decades. ... Read more

Book 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.

Customer Reviews (5)

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.

down to earth book for people like you and me
I have just finished a class using this book, and on the whole its done a good job.I didn't find it in any way super special or anything, but I could read it and understand it.As far as math books go that is pretty good. Lots of exercises with answers in the back, which is what you need.Usually there are worked out examples of the most standard problems, but not always, e.g. there is no example of residue calculus with a Log function. ... Read more

Book 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 (7)

Should be renamed Complex Analysis for Pompous Retards
If you are a really good engineering or science student, sooner or later this book will just really p-ss u off.

In truth, Complex Variables provides a seemingly magical way of dealing with physical situations that can be described with two dimensional partial differential equations (i.e.,conservative fields, conservative forces, and conservative potentials in a two-dimensional plane).Once I got how to apply Complex Variables to physical situations, I could only wish it could be extended to three or more dimensions.Alas, vector calculus appears, to me, the only way to approach 3 dimensional partial differential equations (the only bonus appears that vector calculus can easily handle non-conservative fields, non-conservative forces, and non-conservative potentials).

Most unfortunately, this book doesn't explain the very simple magic of complex variables in physical situations (the whole reason for complex variables unless u r also a mathematician). An astonishingly clear explanation can be found in one of my previous reviews (and I only gave it 3 stars! ...just goes to show u to be sure and take these thoughts with a grain of salt).My efforts would become too dull to other readers were I to reveal exactly which one.:-)

Moreover, a good engineer or a good scientist (physicist) would intuitively expect a fundamental theorem like the Cauchy-Goursat theorem. A book that claims to be for engineers and physicists (this one does in its Preface) should explain what a finely honed engineering or physical intuition knows. The intuition comes from understanding conservation laws and continuity equations.The complex variable Z = X + iY has all the earmarks of a conservative force or a conservative field.Since a good physicist or a good engineer understands how conservative forces work in the real world (gravity and electricity), then he or she already has a fairly strong understanding how they are going to work in the abstract.THERE IS NO MENTION here OF for what reason or for what situations the cauchy-goursat theorem follows from an understanding of the conservative nature of gravity or electricity.I can overlook that in a book aimed at mathematicians, I cannot with one that claims and presumes that it's aimed at engineers.

Anyway, only a retard would not be ultimately offended by this book.It might take a year, two years, or maybe even 10 years.But I firmly believe that sooner or later a good scientist or engineer who understands the other book will become irrately offended by this one.

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).

EXCELLENT BOOK ON COMPLEX ANALYSIS
This is the finest book I have seen in complex analysis. The book that my class was using sucked so bad, I could not get anything from it. It was written in such an advanced manner, as if the student is studying his Phd in math. I thought I'm going to flunk the class, until i purchased this book! This book turned out to be a lifesaver for me. The author does a brilliant job in explaining complex analysis. Anyone who has never ever studied complex analysis and just has a background in basic calculus, will master this field simply by studying this book. Well done job! ... Read more

Book 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. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.

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

Customer Reviews (21)

Rewards you again and again
A few words for the person thinking of buying and using this book.

First, go for it.Don't be scared.But you need to be prepped a little on how it all fits together.Roughly, it breaks up into a course on real analysis (with quite a lot of supplementary material, especially on Fourier analysis), and then breaks into complex analysis in chapter 10.

Now on the first part--You might be tempted to ask "what am I learning?" as you start on the first chapter.It seems like Rudin is taking you the longest way possible.Is he torturing me?, you wonder.Can't you make this more concrete?, you ask.Keep going, and you will begin to see what he's up to.The reason he wrote chapter 1 the way he does is because (note) it involves no structure *except* the space X and the sigma algebra M.He's showing you, in other words, what you can rely on no matter *where* you are, no matter what measure you are using.That taken care of, it's off to chapter 2, where he stirs in a topology T.Two main goals: the Riesz Representation theorem (version 1) and the construction of Lebesgue measure.Try hard to make it through this and see what Rudin is doing.Make some time to read through it, and you will really gain some insight into what Lebesgue measure on R^k does to the study of analysis, extending it beyond Riemann.Personally, I'm still not sure it's 100 percent the best way to do things, but--like I said--it's pretty cool stuff after the hard work is finally over.

Next, he goes through some chapters on different sorts of function spaces that are more or less self contained (all in good time, m'boy), and then he comes back around in Complex Measures (chapter 6) to really hammer in some important things--especially the Radon-Nikodym theorem.Now you're ready to learn about differentiation in chapter 7.By now, you will have seen for yourself what I am talking about--the book really gives it to you straight, and best of all, when you're done learning out of it, it continues to be valuable as a reference because of the meticulous organization of each chapter.

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.

Necessary, Necessary, Necessary
While I would not recommend this text to someone wishing to teach herself real and complex analysis, having this book in your personal mathematics library is a must for anyone seeking to further her education in higher mathematics.It's one of the most commonly used undergraduate texts, referred to by some as the "Bible".If you can afford it though, I would recommend that you pick up a copy of Baby Rudin to use as a reference.

The first two chapters in combination with Bartle's text on Lebesgue Integration and Measure makes for a killer introductory course in Measure Theory.

Oh, and if you can solve the problems in Rudin's book, you can do pretty much anything, so it's a major confidence booster!

Real and Complex Analysis (Higher Mathematics Series)
The approach in this book is formal, yet not intuitive and neither natural for a beginning graduate student who have yet developed some level of mathematical maturity.

Concise and concrete proofs, chanllenging exercises are given in the text. The book is fruitful in many ways, however you must have considerable mathematical maturity in order to benefit from this text.

It is a pleasure to have this book on my shelf. ... Read more

Book Description
This unusually lively textbook on complex variables introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. "Complex Analysis" offers new and interesting motivations for classical results and introduces related topics that do not appear in this form in other texts. Stressing motivation and technique, and complete with exercise sets, this volume may be used both as a basic text and as a reference. For this second edition, the authors have revised some of the existing material and have provided new exercises and solutions. ... Read more

Customer Reviews (8)

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.

Not enough for getting a complete perspective.
My comment refers to the third edition of this book, but I don't think the fourth could be much better.

First of all, this title shouldn't be included in the "Graduate Texts in Mathematics" series becausethe material it covers is covered in introductory undergraduate courses.Second, eventhough the author made a great effort to include as much topicsas he could, the treatment of most of them is highly old-fashioned. I mean,he pays no attention to the most recent and elegant refinements of thebasic theory, so the student is not immediately able to understand the realimportant ideas behind the subject. For example, nowadays the proof of theCauchy integral formula is presented as a more ar less easy corollary ofthe general Stokes theorem. The Cauchy integral theorem is also obtainedeasily following the same fashion. Incredibly, the author explores thisline in one appendix, but not well done, and apparently he doesn't realizethat there is the key idea.

Also, keeping in mind that holomorphicfunctions are harmonic, most of the important results for holomorphicfunctions should follow at once from the corresponding ones for harmonicfunctions, but this old-fashioned texts don't take this remarkableimportant feature of complex analysis into account, making the treatmentinnecessarily complicated and leading the student to misunderstand bothcomplex and harmonic analysis. Eventhough the book includes a whole chapteron harmonic functions, the author doesn't use their power as heshould.

I'm afraid there are few famous introductory texts that I wouldsuggest for first-timers. The best of them is Markushevitch, unfortunatelyout of print.

There is also another serious drawback: The author pays noattention at all to boundary value problems and therefore to theCauchy-type integral, maybe the most important tool of complex analysis.The Hilbert transform is also not present.

If you have the opportunitytake a look at Muskhelishvili's "Singular Integral Equations" andGakhov's "Boundary Value Problems" and then you will understandmy point.

Lang's book could be used as a companion text and as areference for introductory courses. It's got some interestigexcercises.

Its contents are: Complex Nubers and Functions; Power Series;Cauchy's Theorem, First Part; Winding Numbers and Cauchy's Theorem;Applications of Cauchy's Integral Formula; Calculus of Residues; ConformalMappings; Harmonic Functions; Schwartz Reflection; The Riemann MappingTheorem; Analytic Continuation Along Curves; Applications of the MaximumPrinciple and jensen's Formula; Entire and Meromorphic Functions; EllipticFuctions; The Gamma and Zeta Functions; The Prime number Theorem;Appendices.

Please take a look to the rest of my reviews (just click onmy name above). ... Read more

Book 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 (9)

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!)

Outstanding book: very clear, covers a great deal of material too
This is the closest I come to a favourite book on Complex Analysis.It wins on clarity, amount of material covered, and the order in which topics are presented.

Gamelin's writing is very clear and he provides a lot of motivation and discussion; his proofs are easy to follow, and the book has a healthy dose of geometry that clarifies and enriches the subject.In spite of being very easy to read, this book manages to cover a lot of ground, and gets into some more advanced topics.I have not been able to find any book that is as accessible as this book is while also being as comprehensive.Also, this is one of the few books that explores the connections of complex analysis to applied mathematics and pure mathematics equally well.

This book's greatest asset is that it address the differences in background that students inevitably have when approaching the subject of complex analysis.This book covers all the necessary ground thoroughly, but in neat sections which are easy to skip, and it introduces more advanced topics (again in neat sections) very early so that students with a strong background will not be bored.The early introduction to Riemann surfaces is outstanding and greatly enriches the study of the material; the book's final chapter presents the theory rigorously.The two chapters on integration move slowly, but develop the subject in a manner that explores the rich interplay between the theory of analytical functions and the general theory of differentiable functions of two variables.

The exercises are outstanding.They are fairly diverse in difficulty level, and very interesting and informative.They range from simple computations up through interesting tangential theorems.

I think this would make an outstanding text for an undergraduate complex analysis course, and it might make a good text for a graduate course as well.It is also useful for self-study or as a reference.The best part about this book is that it can be used for a first course, and motivated students can plow through the rest of the book, getting into some more advanced and interesting material.After this book one should have no trouble tackling a denser text like the Ahlfors.

One for your reference shelf.
Gamelin's 'Complex Analysis' is purported to be a text that, while it falls in the UTM series, can really be used for anything up through the Ph.D. qualifying exam level.This is true, but there are some problems with this text that would keep it from true brilliance.

The text covers a superb variety of topics, from the basic arithmetic up through graduate level complex analysis.The exercises to be found at the end of each section are likewise excellently chosen, and give students some great 'hands-on' practice using complex analysis.The exercises are often a little too easy, at least early on in the text, and can lull a reader into a great false sense of security with the field.

The approach that Gamelin takes makes for a very readable book, one that can easily give you an idea of what is happening.The problem however is that there is a level of generality--as well as rigor--that are sorely lacking from this text.The results that Gamelin presents can (here and there) be generalized without too much work, which really should be done for a graduate course.Similarly, his writing rather often seems to lack any semblence of the rigor that students of analysis would normally expect.Justifications would be a better word than proofs for many of the ways he convinces a reader a theorem must be true.

This does not detract from the value of the book however, but merely shift it to a different role in one's study of complex analysis.This is a great companion book--one that should find a well worn home on your reference shelf.It is an excellent book to go to when you want to get an idea of what a concept means, and then get a variety of doable problems that relate to that idea.

It's pretty good
The explanations were clear, and the exercises are usefull.It's fairly rigorous, but avoids getting so tangled up in rigor that it obscures the conceptual development of the book.Also, it avoids the persistent (and dreadfull) habit of presenting in the proposition, lemma, corollary, theorem, format which finds its way into a lot of analysis books.Also, I though it was nice that the book develops some concepts from real analysis (continuity, convergence of sequences and series, etc...) so that the book was fairly self contained.Finally, I liked the two sections on applications to fluid dynamics, but I wish the book would have included some further applications.Overall, a good introduction to the subject.(Although it contains way more material than one can cover in a semester) ... Read more

Book 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 graduate course a great deal of flexibility in structuring a more advanced course. This is a revised edition, new examples and exercises have been added, and many minor improvements have been made throughout the text. ... Read more

Customer Reviews (4)

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.

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

Chp 8:Harmonic functions; Chp 9: Schwarz reflection; Chp 10: Riemann mapping theorem; (11):Analytic continuation along curves; (12)applications of Maximum Modulus Principle an Jensen's Formula; (13)Entire& Meromorphic functions; (14) elliptic functions; (15) Gamma & Zetafunctions; (16) The Prime Number Theorem; and a handy appendix. ... Read more

Book Description

One of the most diverse branch of mathematics, complex variables proves enormously valuable for solving problems of heat flow, potential theory, fluid mechanics, electromagnetic theory, aerodynamics and moany others that arise in science and engineering. As taught in this exceptional study guide, which progresses from the algebra and geometry of complex numbers to conformal mapping and its diverse applications, students learn theories, applications and first-rate problem-solving skills.

Customer Reviews (12)

Even in mathematics, Intuition remains the source of truth
This book, although providing solutions and/or answers to all problems, still focuses on a very formal treatment of complex variables.The selection of problems is highly abstract.It remains up to the reader to develop their own sense of direction in the realm of complex numbers.Instead of problems that focus on the final validation of truth in complex numbers, more problems of an applied nature would have been helpful.Even if advanced physics would need to involved.

good reference book...
more of a handbook with the important theorems and formulas.
the examples and excercise are well conceived.

This bookno longer need a review
This book no longer need a review. It is so popular among the academics and the students for its lucid way of treating complex variablesI used this book as my reference for complex variables for the graduate mathematical methods course. This book helped me a lot with lots of examples and interesting exercise problems. It is also verygood for students who wants to have a fast glance at the concepts. Overall, I would strongly recommend this book to any student who wants to learn complex variables in the most simple way with all kinds of examles to solve problems and score high grades.

Great cheap text on complex variables for the mathematician
Complaints seem to abound in regard to how this Schaum's outline is too theoretical and has too few problems involving applications. You must remember that this particular outline was meant to complement an undergraduate mathematics course in complex variables, not an applied physics or engineering course using complex variables. Thus, the purpose of this book is to develop the calculus of functions of a complex variable.

This is one of those Schaum's outlines that has sufficient explanation, figures, and examples that it can double as a cheap textbook on the subject. However, remember that the emphasis is on theorems and proofs of theorems versus applications. However, there are some sections of the outline that are excellent at illustrating some applications of the subject matter. In particular, chapter 9, "Physical Applications of Conformal Mapping" contains applications from physics using those equations that are defined by a potential, including the electromagnetic field, the gravitational field, and, in fluid dynamics, potential flow, which is an approximation to fluid flow assuming constant density, zero viscosity, and irrotational flow. By choosing an appropriate mapping, the outline demonstrates clearly how one can transform the inconvenient geometry of one set of these equations into a much more convenient one. The equations are solved in this new "convenient" geometry, and then transformed back into the old one. One example of a fluid dynamic application of a conformal map that is detailed is the Joukowsky transform.

If you are not looking for a book to complement a mathematics course on complex variables and you are looking for something more applied, you might look at "Complex Variables: Introduction and Applications". That book has the first part dedicated to theory and the second part dedicated to applications at a reasonable price.

too theoretical
I bought this thinking it would help me understand complex variables, complex integration and differentiation. As another customer commented there are wayyyy to few solved concrete problems, all of the solved problems are proving some theorem. This is useless. I can look those proofs up elsewhere. What I expected was concrete solved problems, there are very very few of those. All in all I am rather dissapointed with this book. Not recommended unless you are looking for many proofs of just theorems. ... Read more

Book Description

Customer Reviews (6)

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.

Pay a salute to Dr. Richard Silverman
My aim is not to comment on the book.
I justwant to pay a salute to Dr. Silverman who have spent
his time in translating mnay of the books written not on English.
Most of the books are of high quality.
Without him, we have no chance in reading these books.
Thnaks a lot to Dr. Sliverman again!!
One pity is that in the translating books,
no mention of Dr. Silverman's life.
For example, which university he is teaching?

excellent, rigorous work
I was amazed by this book. In a small amount of space, it manages to present most of the important theoretical aspects of complex analysis, and rigorously, so you get all the detailed proofs. However, the book isn't big on applications, so you might consider getting an applied text tosupplement this one. Also, the book is quite advanced. Some background inadvanced calculus (Widder's book works great) would help you make moresense of the text. I read this after I learned applied compl. analysis, soI can't really judge this book as an introduction to the field, but forsomeone who is familiar with the essentials of complex analysis, this is anexcellent theoretical supplement. ... Read more

Book Description
Complex analysis is one of the most attractive of all the core topics in an undergraduate mathematics course. Its importance to applications means that it can be studied both from a very pure perspective and a very applied perspective. This book takes account of these varying needs and backgrounds and provides a self-study text for students in mathematics, science and engineering. Beginning with a summary of what the student needs to know at the outset, it covers all the topics likely to feature in a first course in the subject, including: complex numbers, differentiation, integration, Cauchy's theorem, and its consequences, Laurent series and the residue theorem, applications of contour integration, conformal mappings, and harmonic functions. A brief final chapter explains the Riemann hypothesis, the most celebrated of all the unsolved problems in mathematics, and ends with a short descriptive account of iteration, Julia sets and the Mandelbrot set. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided. ... Read more

Product Description
This book is based on lectures presented over many years to second and third year mathematics students in the Mathematics Departments at Bedford College, London, and King's College, London, as part of the BSc. and MSci. program. Its aim is to provide a gentle yet rigorous first course on complex analysis.Metric space aspects of the complex plane are discussed in detail, making this text an excellent introduction to metric space theory. The complex exponential and trigonometric functions are defined from first principles and great care is taken to derive their familiar properties. In particular, the appearance of π, in this context, is carefully explained.The central results of the subject, such as Cauchy's Theorem and its immediate corollaries, as well as the theory of singularities and the Residue Theorem are carefully treated while avoiding overly complicated generality. Throughout, the theory is illustrated by examples.A number of relevant results from real analysis are collected, complete with proofs, in an appendix.The approach in this book attempts to soften the impact for the student who may feel less than completely comfortable with the logical but often overly concise presentation of mathematical analysis elsewhere. ... Read more

Customer Reviews (1)

The best I have seen on the subject
This is a great book on complex analysis . It is mathematically well
motivated. So every step is taken for a given reason. The steps donot drop from thin air. It covers a lot of material in a compact and
understandable form and yet is is only 238 pages long. The book has some
surprises I have never seen anywhere. For instance on page page 11 the
author after saying" now watch closely", proceeds to show , without no obvious flaws that 1=-1 !This is carefully explained latter on page 103
immediately after showing an example were [(z.w) ^1/2] is not equal to
[(z)^1/2 . (w)^1/2]. As you can see complex powers are thoroughly explained and so are the other topics treated in this book
... Read more

Book Description
Written for junior-level undergraduate students that are majoring in math,physics,computer science,and electrical engineering. ... Read more

Customer Reviews (2)

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

Book Description

In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations. Subsequent chapters then develop such topics as Hermitian exterior algebra and the Hodge *-operator, harmonic theory on compact manifolds, differential operators on a Kahler manifold, the Hodge decomposition theorem on compact Kahler manifolds, the Hodge-Riemann bilinear relations on Kahler manifolds, Griffiths's period mapping, quadratic transformations, and Kodaira's vanishing and embedding theorems.

Book Description

The idea of this book is to give an extensive description of the classical complex analysis, here ''classical'' means roughly that sheaf theoretical and cohomological methods are omitted.

Customer Reviews (2)

Excellent book/Terrible translation
This book was originally written in German, and the German version is just incredible: a real gem.Good reason to translate it!Unfortunately, this is one of the worst translations from German I have seen.Some of it is just awkward grammar, which the reader may be able to ignore.But, there are also some words and phrases which are translated incorrectly.
For example "Paragraphen" in German does not mean paragraph in English, it means section.But in this book it is translated as paragraph.Try looking for something at the end of a paragraph or in the previous paragraph, when you should actually be looking at the end of the section or in the previous section.An example of this can be found in the explanation of the addition theorem for complex exponents (p. 27).The English text claims there is a remark concerning this at the end of the "paragraph."The paragraph ends and there is no remark.Turn to the end of the section (p. 31) and you will find the remark just above the exercises.
My advice is, if you can read German, get the German version!If you can't read German, you can still get the English version, but you will have to be very patient with the mistakes incurred in the translation (not to be found in the German original). If you own this book, you should systematically go through it and replace "paragraph" everywhere with "section."Most of the other translation mistakes can be figured out by context.

see review
I was truly delighted to find this text. It starts off with ordinary complex analysis at the level of sophomore undergraduate students and proceeds well into graduate-level complex analysis (analytic number theory, elliptic functions, abels theorem, etc). The 'advanced' results are shown using standard methods, so it was a great way for someone who learned the nuts and bolts of contour integration to move into theta functions, the prime number theorem, etc etc...fun stuff. ... Read more

Book Description

Customer Reviews (4)

my favorite intro to complex analysis
This is a masterful treatment of the subject, including power series, integration, the homotopy version of cauchy's theorem, residue calculus, differential equations, harmonic functions, and intro to riemann surfaces.

i confess i do not recognize the book i read from the first review here however, as mine begins with formal power series and has rather little on several complex variables.I have both the 1963 hardback and the recent dover reprint and they are identical, but i have not seen the french version.

a translation of a classic
This elegant little book by Henri Cantan covers both complex functions on one and several variables, and in that way (by the inclusion of several variables) it differs and stands out from most other books on complex variables at the beginning US-graduate level. It is a translation of an original French language version. I can recommend both the original and the translation. It is readable, and the exercises are plenty and excellent. Thanks to Dover, the translation is now readily available and cheap.

Cartan's book starts with complex numbers, power series, and a review of the standard complex functions of one variable, e.g., the exponential, and the complex logarithm. Then follow holomorphic functions, Taylor and Laurent expansions, singularities, Cauchy's theorems, residues, analytic continuation, lots of examples, and beautifully illustrated. Included are also geometric topics, elementary complex geometry, Mobius transformations, automorphisms, transformation groups, differential forms, harmonic and analytic functions, Riemann surfaces, and infinite products, and a brief chapter on conformal mappings.

The book is divided pretty evenly between one and several variables, with the second half being several variables. However each part can be read pretty much independently of the other.

The book in its French edition was published first in the 1950ties, and the first English edition in 1963, and then starting 1995 reprinted by Dover. It is suitable as a text for a course or as a supplement in a standard beginning graduate course in complex function theory. 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, but considerably more elementary we recommend the books by Fisher, Volkovyskii et al, Silverman, Schwerdtfeger, and Flanigan. These books however only cover the case of a single variable. Review by Palle Jorgensen, August 5, 2006.

A good introduction
At first the book appeared to be too difficult. Indeed, it is designed for students with some familiarity with Algebra and Topology. However, the more I read the book, the better it appeared. It's chapter on integration is one of the best I've seen. It doesn't seem to cover much, but the addition of some facts from several variables is in very good taste.

Good and affordable.
Cartan's book on complex analysis is not brand new, but that does not matter as the subject itself has been "stabilized" for a long period of time. It is indeed good to see that this "lovely" book being available at the modest price of [price]. The fact that the author was among the founders of Bourbaki may intimidate some would-be readers, but you don't have to be worried. The treatment of the subject is no pedantic and quite accessible for beginners. ... Read more

Book Description

This book presents complex analysis in one variable in the context of modern mathematics, with clear connections to several complex variables, de Rham theory, real analysis, and other branches of mathematics. Thus, covering spaces are used explicitly in dealing with Cauchy's theorem, real variable methods are illustrated in the Loman-Menchoff theorem and in the corona theorem, and the algebraic structure of the ring of holomorphic functions is studied.

Using the unique position of complex analysis, a field drawing on many disciplines, the book also illustrates powerful mathematical ideas and tools, and requires minimal background material. Cohomological methods are introduced, both in connection with the existence of primitives and in the study of meromorphic functionas on a compact Riemann surface. The proof of Picard's theorem given here illustrates the strong restrictions on holomorphic mappings imposed by curvature conditions.

Customer Reviews (1)

A great way to begin learning Complex Analysis
This book, like each lecture by Narasimhan that I have had the pleasure to hear, exhibits masterful understanding and exposition of the material.His proofs are extremely elegant, and it is always clear where he is going with the material.His writing flows very smoothly, and the addition of exercises to this edition is very helpful in the process of learning to prove things in the various areas covered in the chapter.He also does a great job of allowing one to make the connections between complex analysis and other fields of mathematics, and the notes at the end of each chapter offer interesting insight into the material such as various mathematicians involved in developing the topics and other areas of interest relative to the particular chapter.It is rare that an elementary book on complex analysis goes into further topics like several complex variables and complex manifolds, so if you are interested in learning complex analysis and preparing to extend to further related topics, this is the book for you. ... Read more