Product Description
This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable. It treats several topics in geometric function theory as well as potential theory in the plane. In particular it covers: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. The level of the material is gauged for graduate students. Chapters XIII through XVII have the same prerequisites as the first volume of this text, GTM 11. For the remainder of the text it is assumed that the reader has a knowledge of integration theory and functional analysis. Definitions and theorems are stated clearly and precisely. Also contained in this book is an abundance of exercises of various degrees of difficulty. ... Read more

Product Description
Designed to be used in a one semester course of three credit hours. Among the topics discussed are: complex numbers, integration in the complex plane, laplace transforms and stability of systems, and conformal mapping. DLC: Functions of complex variables. ... Read more

Customer Reviews (10)

TRASH
The other reviews must either be the work of idiots or the publisher of the book.

First of all, this book has NO proofs.It doesn't even have exact definitions and formulations of theorems.

It is a bloated mess - the author doesn't even manage to define what a complex number is.

The only redeeming quality of this book are the frequent references to other books - the author at least now and then must feel guilty about the garbage he is writing, and so he refers you to his betters.

Clear, Concise and most important...INTERESTING
I had the extreme good fortune of having the author as a professor in the 70's and was excited to find this book available.Without a doubt, the best teacher I have ever had, his classroom style comes through vividly in this book.I've read other books on this subject, but none have ever inspired me the way this one has.Real world examples add to the importance and make the math come alive.Examples give very helpful hints at solving the problems and add immensely to the learning experience.I was also very pleased to see a screen shot of Richard Feynman's notebook page where he proclaims e^i(pi)+1=0 as the most remarkable formula in math...indeed!Other historical notes bring the savants of the past to life.You will find that if you have a question about something as you are reading, usually it is magically addressed within the next few sentences.If you enjoy this topic BUY THIS BOOK - you will not be disappointed.

wonderful for self study !!
This book is outstanding for self-study.The discussions are very readable and the problems do a good job of re-enforcing the discussion at hand.The book covers a wide range of topics and spends ample time on each individual subject - so you really get your money's worth. I have made multiple self-study attempts at complex variables using a wide variety of books both old and new and can say that this is the best I've seen.As a bonus Dr. Wunsch takes you on various short historical and mathematical side trips (both in the body of the text and in the problems) which makes the book come alive.I recommend this book - it should set the standard for the definitive undergrad complex variables text book if it hasn't already.

Clearly written with helpful examples
This book is a great way to learn complex variable analysis.The text is generally very clear and concise and there a plenty of good examples.The problems at the end of each section are particularly useful.I agree with other reviews that it is an excellent book to use for self-study or as a reference.I highly recommend it.

A Great Book
My comments here are for the second edition of this book as I do not have the third edition.This is one of the most elegant books on the subject of complex variables.The author uses very clear language to explain complicated topics such as contour integration and conformal mapping.It contains both elementary and advanced topics.This book has several interesting properties:First, It is virtually void of errors.Second, there are a lot of examples worked out in detail. Most of these examples are from physics or engineering.Third,the book contains an excellent collection of homework problems with varying degree of difficulty and the author actually leads you to the solution step by step.This is a great book for self study or any introductory undergraduate course on the subject ... Read more

Product Description
Elliptic functions and Riemann surfaces played an important role in nineteenth-century mathematics. At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations. In this book the authors give elementary accounts of many aspects of classical complex function theory including Möbius transformations, elliptic functions, Riemann surfaces, Fuchsian groups and modular functions. A distinctive feature of their presentation is the way in which they have incorporated into the text many interesting topics from other branches of mathematics. This book is based on lectures given to advanced undergraduates and is well-suited as a textbook for a second course in complex function theory. Professionals will also find it valuable as a straightforward introduction to a subject which is finding widespread application throughout mathematics. ... Read more

Customer Reviews (2)

A postmodern second course in complex analysis
First there are two short chapters on the Riemann sphere and Möbius transformations, probably partly familiar to many readers. Chapter 3 on elliptic functions opens with the following sentence, which illustrates the type of unconvincing pseudo-motivation that occur throughout the book: "Having considered the sphere and its meromorphic functions in the first part of this book, we now turn our attention to another compact surface, the torus, and its meromorphic functions." From here the theory of elliptic functions unwinds along a path that is largely hidden from us, although towards the end of the chapter we are rewarded for fumbling ahead with a discussion of elliptic curves and even a vague allusion to number theory. Now we are warmed up for a general treatment of Riemann surfaces (chapter 4). People often lament that such a simple and beautiful idea requires so much technical machinery to be treated rigourously. Jones & Singerman certainly don't prove them wrong. Chapter 5 is called "PSL(2,R) and its discrete subgroups". This is a heavily geometrical topic, especially since PSL(2,R) is half the isometry group of the half plane model of hyperbolic geometry. Indeed, instead of praising this as a pleasantly geometrical part of function theory, it is perhaps even more satisfying to treat it geometrically altogether (cf. Stillwell). Anyway, the prime example of those "discrete subgroups" of chapter 5 is the modular group, which gets the final chapter 6 all to itself. The whole book has a thoroughly modern feel to it. Admittedly, there is an impressive amount of mathematics for such a modestly sized book, but personally I think there are still many virtues of a more classical and less mysterious approach as well, such as Hurwitz & Courant or Siegel (neither of which is in the bibliography).

invitation to elliptic functions and Riemann surfaces
this book goes from the Riemann sphere up to the modular function and the great Picard Theorem via geometry and group theory; it is an excellent book to begin with from an elementary knowledge of complex functions since it is rather self-contained; I see it as a fine blending of different area of mathematics and as such it should help its reader towards more understanding (through serious work...) of those. As an example, I opened the book at random an found the rule for adding points of a cubic (page 119). As a matter of facts, when I first met hyperbolas as groups via geometric addition of their points, I was rather dumbfounded. It's a pity that the hardback edition cannot be found anymore... ... Read more

Product Description
This textbook introduces the theory of complex variables at undergraduate level. A good collection of problems is provided in the second part of the book. The book is written in a user-friendly style that presents important fundamentals a beginner needs to master the technical details of the subject. Similarly, teachers can also adopt the text for a course on complex variables and for mining problems. The organization of problems into focused sets is an important feature of the book. ... Read more

Product Description

Product Description
Using the same innovative and proven approach that made the authors' Engineering Mathematics a worldwide bestseller, this book can be used in the classroom or as an in-depth self-study guide. Its unique programmed approach patiently presents the mathematics in a step-by-step fashion together with a wealth of worked examples and exercises. It also contains Quizzes, Learning Outcomes, and Can You? checklists that guide readers through each topic and reinforce learning and comprehension. Both students and professionals alike will find this book a very effective learning tool and reference.

Distinctive Features
- Uses a unique programmed approach that takes readers through the mathematics in a step-by-step fashion with a wealth of worked examples and exercises.
- Contains many Quizzes, Learning Outcomes, and Can You? checklists.
- Ideal as a classroom textbook or a self-learning manual. ... Read more

Customer Reviews (1)

"Hard" Math Made Enjoyable & Almost Easy
This is another very excellent and extremely readable book by Stroud & Booth.It's math at its nuts and bolts level. It's geared for engineering types like me, but also for do-it-yourselfers.With Stroud & Booth, there is no such thing as "hard" math. They are uncanny at anticipating those areas where misunderstanding can arise and answering your questions before you even ask them. Every type of problem is not only worked out in detail but is fully explained and fitted into the overall context of the material.You can actually sit down and read this book like a novel and enjoy it.Everything is developed and tied together like the characters and plot of a good novel.For instance, much time is spent bringing you up to speed on partial derivatives, complex mappings and line integrals before the study of Complex Calculus. And, the best part is that you are learning or relearning some very sophisticated math and math tools without pain or boredom.Who would have thought that a former math-hater like me could actually enjoy doing not just line integrals, but Schwarz-Christoffel transformations and other neat stuff.

This book gets through the whole subject nicely. It's concise, precise and yet panoramic.And, once you have mastered Stroud & Booth. you will feel equipped to continue your studies in more traditional texts as I do. The kind that have 800 to 1200 pages of depth and detail on the subject with lots of theory and proofs.

ENJOY! ... Read more

Product Description

The first survey of its kind, written by internationally known, outstanding experts who developed substantial parts of the field. The book contains an introduction written by Remmert, describing the history of the subject, and is very useful to graduate students and researchers in complex analysis, algebraic geometry and differential geometry.

Product Description
For students in mathematics, engineering, and physics.Includes comprehensive coverage of complex numbers, set theory,mapping, functions, Cauchy-Riemann conditions, power series, Taylorseries, Green's theorem, Laurent expansions, singularities, residues,transformations, and numerous scientific applications. ... Read more

Customer Reviews (2)

You need this book for complex variables
Text books on complex variables DO NOT HAVE ENOUGH WORKED PROBLEMS. This 900+ page softcover does. If you study "Visual Complex Analysis" by Needham (one of my favorites) this will help you do the exercises. The 24 chapters cover everything you would find in a one year course and then some. Each chapter is just one solved problem after another. I found a few misprints but nothing serious. You won't find a better resource and cheap too ! Highly recommended.

Another Excellent Resource!
This book is essential for the study of Complex Variables, and it is compatible with other texts on the subject. The sections on how to solve complex contour integrals and Laurent Series are especially important since most texts do not provide enough worked examples to give the student an idea of the underlying patterns involved. The section on the use of Residue theory to solve real valued integrals is also very well done. "Complex Variables Problem Solver" is another fantastic product from the folks at REA. ... Read more

Product Description
Uses introductory problems from particular applications that are easy to understand and show the reader that there is a need for a particular mathematical technique. Numerical techniques are explained from basics with an emphasis on why they work. Discusses the contexts and reasons for selection of each problem and solution method. Worked-out examples are very realistic and not contrived. ... Read more

Customer Reviews (5)

Don't expect to learn from this
If you need a reference for various numerical methods then this book may be of use to you, but do not expect to learn anything from it.You will not be taught 'why' these methods work, or 'how' they came about--it simply lists a bunch of formulas with a few meager attempts at derivations here and there.If you want to actually understand this stuff you'll have to go elsewhere.If you just want a reference...I would still go elsewhere.

Good for reference, but don't expect to learn from this book
I used this book in a college level course, and let me tell you it is close to worthless.I gave it two stars because someone might be able to use this for some kind of whimpy reference book.Don't buy this expectingto learn much from it though.The first few chapters are ok, but later onit just glosses over many things you need in order to understand stuff.

Good Reference Book
This book is exactly as the title says.There are a lot of great sources on applied mathematics and it covers a lot of good material ranging from iterative methods to PDE's.Matlab is the software of choice and the bookdoes a good job of making use of Matlab.The detraction to this book isthe lack of rigor.The book operates under the misconceptiong that appliedmath doesn't have to be very analytical.As a reference book it is goodresource, but as a textbook a little bit more analysis needs to be put in. This is a nice fresh idea for what Numerical Analysis should be, but coulduse a bit more beefing up in the analysis side.Hopefully this will comewith later editions.

Fantastic!Stupendous!
Whether you use Matlab or not, if you need information about numerical methods, this is the book for you.It makes the standard, Press, et al, "Numerical Methods In.." look like a first-grade primer bycomparison.

I'm writing my own book on numerical algorithms for embeddedsystems, so I know whereof I speak._GOOD_ books that are both readable byordinary mortals, and usable for serious computing, can be counted on thefingers of one hand. Most are either too pedantic and obtuse, or too simpleand shallow, or directed at a small subtopic of the larger area ofnumerical methods.Somehow, through a combination of simple, no-snowexamples, lots of exercises, and Matlab programs, author Faucett manages toinclude virtually every aspect of numerical methods.Not only that, butshe includes them in depth, in a way that's easy to follow and exercisesthat drive every point home.In my travels, I've seen a lot of referencesto such things as Runge-Kutta and Adams-Moulton methods, solving stiffODE's, solving for eigenvalues and eigenvectors, and Householderand QRtransformations, to name a few.It's rare to find easily intelligibleexplanations, with derivations, for any _ONE_ of them.To find them all inone book is astonishing.

Clearly, by the title, you get the impressionthat having Matlab will make this book more valuable to you, and it will. However, don't get the idea that the book is only for users of Matlab. Whether you have it or not, you will not find another book so loaded withgems of knowledge about numerical methods.

Fantastic!Stupendous!
Whether you use Matlab or not, if you need information about numerical methods, this is the book for you.It makes the standard, Press, et al, "Numerical Methods In.." look like a first-grade primer bycomparison.

I'm writing my own book on numerical algorithms for embeddedsystems, so I know whereof I speak._GOOD_ books that are both readable byordinary mortals, and usable for serious computing, can be counted on thefingers of one hand. Most are either too pedantic and obtuse, or too simpleand shallow, or directed at a small subtopic of the larger area ofnumerical methods.Somehow, through a combination of simple, no-snowexamples, lots of exercises, and Matlab programs, author Faucett manages toinclude virtually every aspect of numerical methods.Not only that, butshe includes them in depth, in a way that's easy to follow and exercisesthat drive every point home.In my travels, I've seen a lot of referencesto such things as Runge-Kutta and Adams-Moulton methods, solving stiffODE's, solving for eigenvalues and eigenvectors, and Householderand QRtransformations, to name a few.It's rare to find easily intelligibleexplanations, with derivations, for any _ONE_ of them.To find them all inone book is astonishing.

Clearly, by the title, you get the impressionthat having Matlab will make this book more valuable to you, and it will. However, don't get the idea that the book is only for users of Matlab. Whether you have it or not, you will not find another book so loaded withgems of knowledge about numerical methods. ... Read more

Product Description
Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from? In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work. Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains. Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem. This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis. ... Read more

Customer Reviews (1)

Terrific
I give this book high marks as a text for a first course in complex
analysis.Ullrich's presentation is both rigorous and well motivated,
a combination that's very hard to achieve.

Motivation is where this book really shines.For example, in a
number of places, Ullrich pauses in the presentation to remark and
demonstrate how surprising the world of complex-differentiable
functions is, when compared to the world of real-differentiable
functions.In other cases, e.g. in his derivation of Euler's formula
for sin(z), Ullrich shows how an intuitive idea can lead, through
several false starts, to a true result.This is an invaluable
insight, seldom found in textbooks.

The reader should be comfortable with concepts from "Advanced
Calculus" (the real numbers, continuity, convergence, integration,
and so on).Ullrich constructs everything else that's needed.When
the formalism gets heavier, e.g. when he's covering analytic
continuation and the complete analytic function, he introduces it
carefully, explains why it's used, and advises the reader on how
to make sense of it.This makes the book valuable for another
reason -- it works as a fairly gentle introduction to modern,
abstract mathematical techniques.

The book also seems well suited to self-study: it's self-contained
and very clear.
... Read more

Product Description
This work departs from earlier treatments of the subject by emphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, the boundary behavior of holomorphic functions, inner functions, invariant metrics, and mapping theory. While due homage is paid to the more traditional algebraic theory (sheaves, Cousin problems, etc.), the student with a background in real and complex variable theory, harmonic analysis, and differential equations will be most comfortable with this treatment.

It is currently the only book on the subject with exercises and a large number of examples. ... Read more

Customer Reviews (1)

readable and engaging, good introduction to subject
Krantz has a very readable style and this is one math book that is fun reading (assuming you have the background listed above).No definition causes you to wonder why it was defined, and no theorem causes you towonder why it was proved.

It's also one of the few books that definessheaf cohomology in terms of actual geometric intuition and concreteexamples.Even readers not interested in several complex variables shouldbenefit from the way he treats tangential subjects in this book. ... Read more

Product Description
Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples and exercises that illustrate this point. The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $H^p$ spaces and Painlevé's theorem on smoothness to the boundary for conformal maps. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors. ... Read more

Customer Reviews (2)

Not Bad...
This book is reasonably accessible to those who may not have had any previous exposure to complex analysis. Many parts of the text are well written and easy to read and I really enjoyed the exposition on harmonic functions. With that being said, however, there are some things that I did not particularly like.

I thought it was strange that the author discusses the Cauchy integral formula for a disk, develops more aspects of the theory, and then later comes back to deal with homotopy theory and topology insofar as integration is concerned. In this aspect, Conway's treatise on the subject is superior, in my opinion.

I also prefer Conway's proof of Mittag-Leffler's theorem which is eloquent and a good application of Runge's Theorem. Additionally, I prefer Conway's proof of the Picard theorems as well (Conway uses Montel-Caratheodory which in and of itself is interesting while Greene and Krantz use the modular function and there are a few choice spots when Krantz is a bit vague).

Finally, some of the proofs and exercises contain errors (most of them minor, some of them not so minor) and a few of the proofs are quite difficult to follow at times while Conway's book seems more readable in these areas. This comment mainly applies to the 2nd edition and it is quite conceivable that the author has remedied these errors in the 3rd edition.

Overall, this book has some value. I believe that this book, coupled with Conway's book is a good combination. There are many things that Greene and Krantz do that I prefer over Conway and vice versa although if I had to compare the two, I would prefer Conway.

Very clear prose, explanations, good motivation, an all-around well-written book
This book is rather unorthodox in a number of respects, but it has become one of my favourite texts in complex analysis.The authors claim that their motivation for their presentation of the subject is to emphasize the interconnectedness of complex function theory with multivariable calculus, and de-emphasize the connection with topology.While I do not exactly agree with these goals, I think they do an excellent job of acheiving them.My only complaint about the book is that a few proofs in early chapters result in a sea of differential operators that is resolved by a plug-and-chug computation, something I'd always rather avoid.

The level of the book is elementary, especially for a graduate text, and I appreciate the authors for making honest and reasonable claims about the accessibility of their book.This book would probably even work well for someone who has not had a prior course in complex analysis, such as senior undergraduates.Some of the more advanced topics are presented in clearer ways in this book than I have seen elsewhere.

This book has a wealth of exercises, and the difficulty level is somewhat inconsistent.Some of the exercises are outright inane--possibly inappropriate for a graduate-level text, but useful for rote practice.Others are more interesting.I appreciate, however, the inclusion of more elementary exercises: many graduate texts have the problem of not including enough such exercises, which can make it hard for students to master the fundamentals.This book avoids this pitfall.

The best part about this book is the prose.This book is well-written and is a pleasure to read.Theorems and results are well-motivated, and necessary nuances are effectively communicated through the text.The authors do not over-emphasize equations: they use words in a proof when they are clearer.

The book is well-indexed and comprehensive.A student with prior background in Complex analysis might want to read the first chapter to get familiar with the authors' incessant use of differential operators, and then feel free to skip around.I think this book would make an excellent textbook for a course in complex analysis, although the unusual development in the first chapter might annoy some professors. ... Read more

Product Description
From the algebraic properties of a complete number field, to the analytic properties imposed by the Cauchy integral formula, to the geometric qualities originating from conformality, Complex Variables: A Physical Approach with Applications and MATLAB explores all facets of this subject, with particular emphasis on using theory in practice.

The first five chapters encompass the core material of the book. These chapters cover fundamental concepts, holomorphic and harmonic functions, Cauchy theory and its applications, and isolated singularities. Subsequent chapters discuss the argument principle, geometric theory, and conformal mapping, followed by a more advanced discussion of harmonic functions. The author also presents a detailed glimpse of how complex variables are used in the real world, with chapters on Fourier and Laplace transforms as well as partial differential equations and boundary value problems. The final chapter explores computer tools, including Mathematica®, Maple™, and MATLAB®, that can be employed to study complex variables. Each chapter contains physical applications drawing from the areas of physics and engineering.

Offering new directions for further learning, this text provides modern students with a powerful toolkit for future work in the mathematical sciences. ... Read more

Product Description
Presents a comprehensive, detailed, and organized treatment of the foundations of a complex variable. Covers everything from the foundational work fo Fatou and Julia, to the most recent results, such as those of Sullivan and Shishikura. Softcover. DLC: Iterative Methods (Mathematics). ... Read more

Customer Reviews (1)

Pictures and math.
Many people outside mathematics know of Julia sets, Mandelbrot sets, attractors, chaos, and fractals. They might have encountered them in color renditions as moving pictures that serve as computer screen savers. The book is about the underlying mathematics.

The rational functions (i.e., quotients of polynomials) form a relatively simple class of functions of a single complex variable. One of the operations we do on functions f is substitution: Starting with a complex number z (i.e., a point in the plane), repeated application of f yields a sequence of complex numbers (points), z, f(z), f(f(z)), etc. The resulting sets have a surprisingly rich structure, and they serve as models for dynamics that arises in different subjects. They are the subject of this book. I will let the reader discover the precise definitions in the book: The sets are defined from some basic geometric and analytic properties of these sequences. The set of points z where the sequence has one of two exclusive analytic properties divides the plane into two parts, an `inside' and an `outside', if you like. Well, the mathematical definition is more complicated than that, but the idea is roughly as I said.

The French mathematician Gaston Julia (1893-1978) was the first to make precise the sets that we now know from a host of fun computer experiments. And it is interesting that Julia invented the sets in 1919 without the benefit of computer programs.

Indeed the subject stayed in the background in the mathematical fashions until it became possible to easily visualize the possibilities with simple computer experiments. As a result, the interest in Julia's work had a more renaissance, and its connections to exciting properties in non-linear dynamics turned into an exciting subject in pure and applied mathematics.

This beautifully written book is primarily about the mathematical side of the subject, but it is full of hands-on-examples that illustrate the theory, and that are easy to follow for students. The exercises are lovely, and make the book eminently attractive for the teaching of the subject to students who will need very little more in the way of background than a familiarity with the complex numbers.
Palle Jorgensen, October 2004.
... Read more

Product Description

This book gives a comprehensive introduction to complex analysis in several variables. It clearly focusses on special topics in complex analysis rather than trying to encompass as much material as possible. Many cross-references to other parts of mathematics, such as functional analysis or algebras, are pointed out in order to broaden the view and the understanding of the chosen topics. A major focus is extension phenomena alien to the one-dimensional theory, which are expressed in the famous Hartog's Kugelsatz, the theorem of Cartan-Thullen, and Bochner's theorem.

The book primarily aims at students starting to work in the field of complex analysis in several variables and teachers who want to prepare a course. To that end, a lot of examples and supporting exercises are inserted throughout the text, which will help students to become acquainted with the subject.

Product Description

The authors’ aim here is to present a precise and concise treatment of those parts of complex analysis that should be familiar to every research mathematician. They follow a path in the tradition of Ahlfors and Bers by dedicating the book to a very precise goal: the statement and proof of the Fundamental Theorem for functions of one complex variable. They discuss the many equivalent ways of understanding the concept of analyticity, and offer a leisure exploration of interesting consequences and applications. Readers should have had undergraduate courses in advanced calculus, linear algebra, and some abstract algebra. No background in complex analysis is required.

Customer Reviews (1)

A tribute to old masters
the main feature of this book is inscribed in its title:"in the spirit of Lipman Bers"; so, it is a testimony to their (the authors) beloved teacher; as a matter of fact, in the bibliography they point to one of Bers's prefered author: Konrad Knopp: he even translated one of Knoppp's five books on function theory (Funktionentheorie in german), adding that whenever Bers had to advise about exercice books on this subject, he told his students to work in those books. As a tribute to their past master, I must award 5 starsto the book's authors, but something has been nagging at me since I first opened the book; first of all, there is a chapter on the topology of H(D), i.e. the topological vector space of all holomorphic functions on an open Subset D of the complex plane (the topology being the so called "topology of uniform convergence on compact subsets", which is a fine idea, albeit it is taken from a to z in Cartan's book: "Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes"which has been translated in english as: "Elementary Theory of Analytic Functions of one or Several Complex Variables" (Dover editions); it is a sad thing to observe that no tribute at all is given to the old gentleman (Cartan is currently 104 years old...) although his book is indicated in the bibliography with the single mention: "begins with a treatment of formal power series..."; as a matter of facts, I feel that Cartan's choise of exposition is both straitforward and flawless: you get power series and the exponential function (together with pi's definition, complex logarithm and complex trigonometry) in 30 pages, Goursat's theorem and Cauchy's integral form within 70 pages (marginally better than in the currently reviewed book); this is my first point; then, towards the end of the book, one can find a proof of the celebrated Stirling formula; this time, it is taken from top to bottom out of Ahlfors's book: "complex Analysis", truly as magnificent a book as Cartan's and this time they aknowledge it; all the better..., I think that Ahlfors's book is really a great book too. Bluntly, the book contains less material than Cartan, is much more expensive (but less than Ahlfors) so why bother with it? On the other side, the authors have taken stock from the some of greatest specialists and writers in the field (from a classical point of view) which means that their work contains some outstanding mathematics. In the end, it is four stars as a tribute to Bers, Cartan, Ahlfors and Serge Lang who is said to have traded a lot of ideas with Bers while he was writing his own book on complex analysis
(guess what: from page37 to 46, Lang begins his chapter on power series
with formal power series (shades of his french background?). ... Read more

Product Description

This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language. The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. ". . . far more readable than most . . . it is also much more complete." Olivier Debarre in Mathematical Reviews, 1994.