An Introduction to Complex Analysis in Several Variables (North-Holland Mathematical Library)

Book Description

A number of monographs of various aspects of complex analysis in several variables have appeared since the first version of this book was published, but none of them uses the analytic techniques based on the solution of the Neumann Problem as the main tool.

The additions made in this third, revised edition place additional stress on results where these methods are particularly important. Thus, a section has been added presenting Ehrenpreis' ``fundamental principle'' in full. The local arguments in this section are closely related to the proof of the coherence of the sheaf of germs of functions vanishing on an analytic set. Also added is a discussion of the theorem of Siu on the Lelong numbers of plurisubharmonic functions. Since the L ² techniques are essential in the proof and plurisubharmonic functions play such an important role in this book, it seems natural to discuss their main singularities.

Customer Reviews:

High-level math, as expected from Hormander.......2000-04-14

I started to learn several complex variables a few weeks ago, and I noticed the absolute lack of textbooks on the subject. Probably the book that comes more naturally as an extension of undergraduate complex analysis is Gunning and Rossi, but this title is out-of-print (even finding a used copy is nearly impossible. Believe me, I've tried hard).

So, we have Hormander's book. Lars Hormander is known for writing high-level math texts (both in quality and difficulty), as seen in his famous 4-volume series about PDE's, and this book is no exception. His point of view is more related to his area of research (PDE's, again), and his demands for prerequisites are higher than GR (basics from Lebesgue integration, differential forms, algebra and point-set topology are more than welcome), but this book is a masterpiece of mathematical craftsmanship. The methods here developed are often unique, and the author presents the subject in a fully rigorous way. Along with the fact that it is one of the very few books on several complex variables still in print, this is a very valuable text, set in a high standard of excellence. My only complaint is the obscenely high price for a book so important. Several complex variables are an indispensable background for complex manifolds and algebraic geometry, and several important topics in theoretical physics (string theory, twistor theory, conformal field theory), and it's a shame that books like GR go out-of-print without any others for substituting them. Hormander's book doesn't go much deep in these directions, but you won't find any other book in print on the subject with such a high quality.

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An Introduction to Complex Analysis
O. Carruth McGehee
Manufacturer: Wiley-Interscience
ProductGroup: Book
Binding: Hardcover

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ASIN: 047133233X

Book Description

Recent decades have seen profound changes in the way we understand complex analysis. This new work presents a much-needed modern treatment of the subject, incorporating the latest developments and providing a rigorous yet accessible introduction to the concepts and proofs of this fundamental branch of mathematics. With its thorough review of the prerequisites and well-balanced mix of theory and practice, this book will appeal both to readers interested in pursuing advanced topics as well as those wishing to explore the many applications of complex analysis to engineering and the physical sciences.
* Reviews the necessary calculus, bringing readers quickly up to speed on the material
* Illustrates the theory, techniques, and reasoning through the use of short proofs and many examples
* Demystifies complex versus real differentiability for functions from the plane to the plane
* Develops Cauchy's Theorem, presenting the powerful and easy-to-use winding-number version
* Contains over 100 sophisticated graphics to provide helpful examples and reinforce important concepts

Introduction to Riemann Surfaces (CHEL/313)

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Introduction to Riemann Surfaces (CHEL/313)
George Springer
Manufacturer: Chelsea Publishing Company
ProductGroup: Book
Binding: Hardcover

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ASIN: 0828403139

Book Description

This well-known book is a self-contained treatment of the classical theory of abstract Riemann surfaces. The first five chapters cover the requisite function theory and topology for Riemann surfaces. The second five chapters cover differentials and uniformization. For compact Riemann surfaces, there are clear treatments of divisors, Weierstrass points, the Riemann-Roch theorem and other important topics.

Book Description

Complex variables offer very efficient methods for attacking many difficult problems, and it is the aim of this book to offer a thorough review of these methods and their applications. Part I is an introduction to the subject, including residue calculus and transform methods. Part II advances to conformal mappings, and the study of Riemann-Hilbert problems. An extensive array of examples and exercises are included. This new edition has been improved throughout and is ideal for use in introductory undergraduate and graduate level courses in complex variables. First Edition Hb (1997): 0-521-48058-2 First Edition Pb (1997): 0-521-48523-1

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Complex variables provide powerful methods for attacking problems that can be very difficult to solve in any other way, and it is the aim of this book to provide a thorough grounding in these methods and their application. Part I of this text provides an introduction to the subject, including analytic functions, integration, series, and residue calculus and also includes transform methods, ODEs in the complex plane, and numerical methods. Part II contains conformal mappings, asymptotic expansions, and the study of Riemann-Hilbert problems. The authors provide an extensive array of applications, illustrative examples and homework exercises. This new edition has been improved throughout and is ideal for use in undergraduate and introductory graduate level courses in complex variables.

Customer Reviews:

Excellent book -- Unique selection of topics.......2003-03-08

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

A very good text!.......2002-11-30

A very good text!

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

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

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

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

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

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

Highly recommended!

The numerous pictures are enough to recommend this text.......2001-06-07

This text is distinguished by the numerous diagrams that appear on practically every other page. If you're graphically oriented, like I am, then this itself is enough to recommend this book. Concepts such as branch points and multivalued complex functions are much easier to understand when there is a picture to accompany the concept. The second half of the book is concerned with applications and includes several useful asymptotic methods such as Laplace's integral method. These asymptotic techniques are good for evaluating particularly nasty integrals in which the integrand is really concentrated somewhere in the interval. On the downside, this is not a very formally rigorous book. On the other hand, such formalism is easier to digest once you've seen numerous pictures and examples, in my own opinion.

A complete reference book for complex variables.......2000-12-11

Whether you are a student, or just in need of a good reference text, Mark J. Ablowitz' and Athanassios Fokas' book belongs in your library. Complex Variables, Introduction and Applications is refreshingly well written. In clear and logical flow, the authors present the subject of complex variables in an easy-to-understand, yet complete format suitable for both students and practicing professionals.

This text offers a broad coverage of the subject, from fundamental properties of complex numbers, analytic functions, and singularities to more advanced topics such as conformal mapping and Riemann-Hilbert problems. Although individuals interested in pure mathematics may find some of the proofs insufficiently rigorous, those using the book as a reference for engineering or scientific problems may find the text too rigid. Overall, however, the authors have done an excellent job balancing the subject matter and successfully achieving their goal of, when necessary, "sacrificing a rigorous axiomatic development with a logical development based upon suitable assumptions."

Although the mathematical development of the text is clear, concise, and easy to follow, many of the applied examples, such as those for uniform flow in section 2.1, would benefit from further physical insight. Individuals already familiar with physics will have no difficulty following many of the examples, and extending them to other situations. Those less grounded in the physical sciences, however, may find the starting equations for some of the examples to be less than intuitive. Though additional explanations would increase the book's already substantial heft, the change would benefit many readers.

It is a joy to read a well-written technical book with almost no typographical or technical errors. Except for minor (and easily recognizable) typographical errors such as that in equation 2.2.12b, the book is nearly flawless. This leaves the student free to concentrate on learning the material unencumbered by worries about the text's accuracy.

The index is nicely composed, complete, and accurate. This makes the book particularly useful as a reference. Typically, the reader will have little trouble using the index to go directly to the pages of most interest and applicability regarding the subject of inquiry. It would be nice to see a more complete bibliography, as well as a summary of common symbols. Especially useful would be a summary of some of the more important equations (such as Green's theorem, Cauchy's theorem, the Fourier transform, the Helmholtz equation, etc.) derived or demonstrated in the book. A list of important equations, particularly, would improve the book's utilization as a desk reference.

For the student, the text presents answers to odd-numbered questions in the back of the book. For the most part, the text presents only the answers, but occasionally the authors provide additional insight into the problem's solution, as in section 5.2. This will be useful for those engaged in independent study.

Overall, this is an excellent text, and one of the most complete and well-written books on complex variables I have seen. I highly recommend it to anyone interested in the subject, and have placed it prominently upon my reference bookshelf.

Definitely a great book!.......1999-11-30

This is DEFINITELY great book! no question about it! it covers everything, it has enough examples, it is very clear, it suits for self learning for undergrate and graduate students, and I definitely recommend it with all my heart.

Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory : De Gruyter Studies in Mathematics Number Three (De Gruyter Studies in Mathematics)

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Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory : De Gruyter Studies in Mathematics Number Three (De Gruyter Studies in Mathematics)
Ludger Kaup , and Burchard Kaup
Manufacturer: Walter De Gruyter Inc
ProductGroup: Book
Binding: Hardcover

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ASIN: 3110041502

An Introduction to Complex Function Theory (Undergraduate Texts in Mathematics)

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Fantastic for self study
Good Secondary Text
Wordy, but comprehensive
Very readable text.
outstanding: a book to be read and reread.

An Introduction to Complex Function Theory (Undergraduate Texts in Mathematics)
Bruce P. Palka
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 038797427X

Book Description

This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a "short course" in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature.

Customer Reviews:

Fantastic for self study.......2007-10-09

I agree that the exposition and proof's are both wordy, but for self study I found this invaluable. I took this course as a reading course, which means no lecture accompanies the course. I find most weeks, I can solve nearly all of the problems assigned by 1-3 readings of the chapter. This is in my opinion the best book I found to date for self study. However, the addition of solutions to selected exercises would be even better. I recommend the book for those wishing for a introduction to complex analysis, or those with some background and wishing to extend their background to include the material covered on most complex qualifying exams.

Good Secondary Text.......2002-11-30

This is a good text that covers somewhat more ground than either Ahlfors or vol. 1 of Conway, including a really nice, long section on conformal mapping. There are an astonishingly large number of exercises. The exposition is generally quite good, and the proofs are thorough and very detailed.

However, I would not want to use this as a primary text because it is too long-winded. Even the proofs have a lot of unnecessary verbiage. Put simply, it takes too damned long to read. I think the happy medium lies somewhere between the slick austerity of Rudin's analysis books and the overwritten style of Palka's book.

That said, I still think Palka's book is well worth having. You may never want to read it cover to cover, but I find it very useful to turn to when I am confused by other texts, or when I want a more detailed explanation of a given topic. Chances are that I'll find it here. Also, there are lots of good concrete examples, especially in the sections on residue techniques for integration and conformal mapping. And you will never run out of interesting exercises if you have this book around.

So what is the best primary text for complex analysis? Good question, and I don't have a good answer. Conway's book is pretty nice but unexciting. Ahlfors' is generally good but requires extremely careful reading (sometimes between the lines) to catch all the nuances; it also feels a bit old-fashioned for my tastes. The recent version of Narasimhan's book, with the large collection of expository exercises by Nievergelt, is much more efficient, sophisticated, and challenging than the others and looks like it will serve as a very good reference/second course after learning the material elsewhere.

Wordy, but comprehensive.......2002-06-20

I took this class from Dr. Palka himself back in 1994. He is obsessed with words when he speaks and naturally it comes across in his text. One of the few math books where I've occasionally been forced to utilize an English language dictionary. I am not sure whether this is a negative or a positive. It certainly didn't result in me learning any less about the subject.

Besides that Palka includes numerous calculational examples and problems covering a lot of extra material that is both useful and interesting. The lack of solutions makes it difficult for self-study, but you learn more from the problems if it's left up to you to be the final authority on whether it's right or wrong.

Anyway, I give the book a 9 for Comprehensiveness and Readability and a 6 for Succintness. But it's definitely a 10 for range and depth of problems and examples.

Very readable text........2001-06-01

I used this book for a first course in complex analysis. This book comprehensively covers the standard topics in a first course. There are also enrichment sections for those who are interested (such as proving certain definitions are equivalent to the usual definitions). The style of writing is very readable, but this is at the expense of using a lot of words and hence the author sometimes takes a long time to explain a simple idea. This book is not written concisely for this reason. Also, the book does not set out definitions as separate paragraphs nor are they numbered (this seems to be increasingly common for math texts); they are buried within the text, and are very difficult to find later on. Proofs are given out in full and explained in detail with lots of words (at the expense of length and terseness), so that readers can understand very easily. There are plenty of examples, and they demonstrate good techniques for evaluating integrals. There are many exercises and solutions are not provided. Historical facts and footnotes are seldomly found. I believe this book is also suitable for less-prepared students.

outstanding: a book to be read and reread........1998-08-14

The best introductory book on Complex Functions I have ever seen. Clear, well-thought, well written, full of examples, leaves nothing out, even some umpleasant facts about Homology and the Cauchy Integral Formula. Many exercises, from easy to challenging, many diagrams, this book leaves the reader ready for advanced work in Complex Analysis. A masterpiece.Gustavo Rocha da Silva.

Introduction to Complex Analysis in Several Variables

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Introduction to Complex Analysis in Several Variables
Volker Scheidemann
Manufacturer: Birkhauser
ProductGroup: Book
Binding: Paperback

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ASIN: 376437490X

Book 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.

An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra (Cambridge Studies in Advanced Mathematics)

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brief, but good

An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra (Cambridge Studies in Advanced Mathematics)
Christian Peskine
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0521480728

Book Description

In this introduction to commutative algebra, the author choses a route that leads the reader through the essential ideas, without getting embroiled in technicalities. He takes the reader quickly to the fundamentals of complex projective geometry, requiring only a basic knowledge of linear and multilinear algebra and some elementary group theory. The author divides the book into three parts. In the first, he develops the general theory of noetherian rings and modules. He includes a certain amount of homological algebra, and he emphasizes rings and modules of fractions as preparation for working with sheaves. In the second part, he discusses polynomial rings in several variables with coefficients in the field of complex numbers. After Noether's normalization lemma and Hilbert's Nullstellensatz, the author introduces affine complex schemes and their morphisms; he then proves Zariski's main theorem and Chevalley's semi-continuity theorem. Finally, the author's detailed study of Weil and Cartier divisors provides a solid background for modern intersection theory. This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra.

Customer Reviews:

brief, but good.......2007-02-21

Peskine doesn't give a lot of explanations (he manages to cover on 30 pages what usually takes up half a book) and the exercises are tough, but the book is nevertheless well written, which makes it pretty easy to read and understand. Recommended for everyone willing to work their way through his one-line proofs ("Obvious.")!

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