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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Vortices in the Magnetic Ginzburg-Landau Model (Progress in Nonlinear Differential Equations and Their Applications)
Etienne Sandier , and
Sylvia Serfaty
Manufacturer: Birkhäuser Boston
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ASIN: 0817643168 |
Book Description
With the discovery of type-II superconductivity by Abrikosov, the prediction of vortex lattices, and their experimental observation, quantized vortices have become a central object of study in superconductivity, superfluidity, and Bose--Einstein condensation. This book presents the mathematics of superconducting vortices in the framework of the acclaimed two-dimensional Ginzburg-Landau model, with or without magnetic field, and in the limit of a large Ginzburg-Landau parameter, kappa.
This text presents complete and mathematically rigorous versions of both results either already known by physicists or applied mathematicians, or entirely new. It begins by introducing mathematical tools such as the vortex balls construction and Jacobian estimates. Among the applications presented are: the determination of the vortex densities and vortex locations for energy minimizers in a wide range of regimes of applied fields, the precise expansion of the so-called first critical field in a bounded domain, the existence of branches of solutions with given numbers of vortices, and the derivation of a criticality condition for vortex densities of non-minimizing solutions. Thus, this book retraces in an almost entirely self-contained way many results that are scattered in series of articles, while containing a number of previously unpublished results as well.
The book also provides a list of open problems and a guide to the increasingly diverse mathematical literature on Ginzburg--Landau related topics. It will benefit both pure and applied mathematicians, physicists, and graduate students having either an introductory or an advanced knowledge of the subject.
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A Primer of Real Analytic Functions, Second Edition
Steven G. Krantz , and
Harold R. Parks
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The Implicit Function Theorem: History, Theory, and Applications
ASIN: 0817642641 |
Book Description
The subject of real analytic functions is one of the oldest in modern mathematics and is the wellspring of the theory of analysis, both real and complex. To date, there is no comprehensive book on the subject, yet the tools of the theory are widely used by mathematicians today. Key topics in the theory of real analytic functions that are covered in this text and are rather difficult to pry out of the literature include: the real analytic implicit function theorem, resolution of singularities, the FBI transform, semi-analytic sets, Faà di Bruno's formula and its applications, zero sets of real analytic functions, Lojaciewicz's theorem, Puiseaux's theorem. New to this second edition are such topics as: * A more revised and comprehensive treatment of the Faà di Bruno formula * An alternative treatment of the implicit function theorem * Topologies on the space of real analytic functions * The Weierstrass Preparation Theorem This well organized and clearly written advanced textbook introduces students to real analytic functions of one or more real variables in a systematic fashion. The first part focuses on elementary properties and classical topics and the second part is devoted to more difficult topics. Many historical remarks, examples, references and an excellent index should encourage student and researcher alike to further study this valuable and exciting theory.
Book Description
Published by McGraw-Hill since its first edition in 1941, this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics. It will primarily be used by students with a background in ordinary differential equations and advanced calculus. There are two main objectives of this text. The first is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. The second is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations.
Customer Reviews:
Excellent.......2003-11-08
This book is quite thorough, but remains easy to follow (considering the material). It starts out with partial differential equations (no previous PDE experience needed) and shows where Fourier series comes from, which I found motivating since the purpose of Fourier Analysis was evident from the beginning. It then goes into making solutions of arbitrary functions out of sine and cosine functions as well as touching on other orthogonal sets.
The book's main focus is on starting with PDEs and ending with a solution of a Fourier series.
The first chapter was the hardest since the approaches to problems were much different than in calculus, but after adjusting to the material and the approaches to the problems, it gets easier!
Great text for an intro to pde's course!.......2003-04-30
My first encounter with partial differential equations was out of this book. Since then, I've had another course on pde's, and used this book as a reference quite often. Fourier Series adn Boundary Value Problems is very much like Complex Variables and Applicatoins, also by Churchill and Brown. It's accessible to a large audience. Though it would help to have had an advanced calculus course, it isn't necessary to understand the mechanics of solving pde's (namely the variables seperable cases, which is mostly what's in this book). If you're an undergraduate math, engineering or physics student, you'll probably be using this book.
Try Another Text.......2000-08-23
I found Dr. Brown, in conjunction with Dr. Churchill, to have written a very dry and non-useful text. It fails to provide the undergraduate student with the resources and background information that more highly touted books offer. There are a few examples that are somewhat helpful, but overall I found myself having to use reference texts to supplement this one. I am not a math major, but am continually searching for good math texts to help me grasp the fundamentals of more difficult topics. I did not find that help here. Too much 'math prose' and not enough to-the-point definitions and examples, which is the cry of every non-math major. Their treatment of the Laplacian is not even worth the bother of placing it in the book. The physical size of the book is small, (9 1/2 by 6") with 335 pages. Not nearly enough for the treatment of its titled subject.
An excellent book on Fourier Series.......2000-02-12
This is a great book that gives precise examples which are easy to comprehend. Dr. Brown proves to be an excellent author once again.
Book Description
Vol. I of Lars Hörmander's 4-volume treatise was an exposition of the theory of distributions and Fourier analysis preparing for the study of linear partial differential operators.
The present Vol. II is mainly devoted to operators with constant coefficients. An analysis of the existence and regularity of (fundamental) solutions in the first two chapters is followed by a thorough study of the Cauchy problem. One chapter is devoted to the spectral theory of short range perturbations of operators with constant coefficients, and another presents Fourier-Laplace representations of solutions of homogeneous differential equations with constant coefficients. The last chapter is a study of the closely related subject of convolution operators.
Book Description
This reader-friendly book presents traditional material using a modern approach that invites the use of technology. Abundant exercises, examples, and graphics make it a comprehensive and visually appealing resource. Chapter topics include complex numbers and functions, analytic functions, complex integration, complex series, residues: applications and theory, conformal mapping, partial differential equations: methods and applications, transform methods, and partial differential equations in polar and spherical coordinates. For engineers and physicists in need of a quick reference tool.
Customer Reviews:
A Very well written text.......2004-08-27
This book is by far one of the best out there in the area of complex analysis. The author combined application with detailed proofs, giving the reader everything that he/she needs. A course with emphasis on applications can be easily planned out using this as the text. Similarly, if an instructor decides to focus more on proofs, the structure of this book allows this too. Simply put, this is the best out there.
You might also be wondering what I think of Churchill's text, as compared to this one. Yes I agree that it is also a good book. However Churchill's proofs are sometimes too hand-waving. Occassionally, I find Churchill using terms like "see exercise for details of this proof". Now, I really dislike this when author does this. This book by Asmar, rarely makes such comment and this therefore is also one of the reasons that I like about this book.
So trust me, if you want to learn about complex analysis, GET THIS BOOK!! Besides, the 2nd half the book talks about pdf, if you are an engineer and needs to deal with partial differential equations at work, this is a 2-IN-ONE text!!!!!
The book is worth every penny that I paid for!! Asmar ROCKS!!
Book Description
Vol. 1 of Lars Hörmander's influential 4-volume treatise is a detailed exposition of the theory of distributions.
From the reviews:
"In order to illustrate the richness of the book: in my review of the 1983 edition [...] I gave a list of 20 subjects which were new compared to Hörmander's book of 1963. Most of these subjects concern important, basic and highly nontrivial theorems in analysis. Hörmander's treatment of these is extremely clear and efficient and often highly original. [...] Most of the exercises are witty, with an interesting point. The phrasing of both the exercises and the answers and hints is very careful [...] In all, the book can be highly recommended, both as a textbook for advanced students, and as background and reference for introductory courses on distributions and Fourier analysis."
J.J. Duistermaat in Mededelingen van het Wiskundig Genootschap
Customer Reviews:
the definitive guide to micro-local analysis.......2000-04-05
Hormander's four volume guide to linear partial differential operators is the definitive guide to smooth micro-local analysis from the man who invented a large fraction of it.
Micro-local analysis is an approach to the analysis of singularities of variable coefficient linear partial differential equations based on the use of the Fourier transform and of symplectic geometry. Ideas from Hamiltonian physics and quantum theory also feature heavily. The fact that the subject is a mixture of analysis and geometry makes it beautiful and powerful.
The four volume set is hard work and in some ways is more useful as a reference work than as a learning book but on the other hand, all the material is there and the books do reward the reader who is willing to spend the time to study them carefully.
The first volume is the most accessible and the second paperback edition carries the added benefit of included exercises with partial answers which are often highly informative.
Average customer rating:
- The best I have seen on the subject
|
Lecture Notes on Complex Analysis
Ivan Francis Wilde
Manufacturer: World Scientific Publishing Company
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Real Analysis: Theory of Measure And Integration
ASIN: 1860946429 |
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.
Customer Reviews:
The best I have seen on the subject.......2007-03-18
This is a great book on complex analysis . It is mathematically well
motivated. So every step is taken for a given reason. The steps do not 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
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Partial Differential Equations and Complex Analysis (Studies in Advanced Mathematics)
Steven G. Krantz
Manufacturer: CRC
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Geometric Function Theory: Explorations in Complex Analysis (Cornerstones)
ASIN: 0849371554 |
Book Description
Ever since the groundbreaking work of J.J. Kohn in the early 1960s, there has been a significant interaction between the theory of partial differential equations and the function theory of several complex variables. Partial Differential Equations and Complex Analysis explores the background and plumbs the depths of this symbiosis. The book is an excellent introduction to a variety of topics and presents many of the basic elements of linear partial differential equations in the context of how they are applied to the study of complex analysis. The author treats the Dirichlet and Neumann problems for elliptic equations and the related Schauder regularity theory, and examines how those results apply to the boundary regularity of biholomorphic mappings. He studies the ?-Neumann problem, then considers applications to the complex function theory of several variables and to the Bergman projection.
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Boundary Behaviour of Conformal Maps (Grundlehren der mathematischen Wissenschaften)
Christian Pommerenke
Manufacturer: Springer
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ASIN: 3540547517 |
Book Description
There has been a great deal of recent interest in the boundary behaviour of conformal maps of the unit disk onto plane domains. In classical applications of conformal maps, the boundary tended to be smooth. This is not the case in many modern applications (e.g. for Julia sets). The first chapters present basic material and are also of interest for people who use conformal mapping as a tool. The later chapters deal in greater detail with classical material and, go into recent developments (e.g. by Makarov). The reader is assumed to know standard complex and real analysis. The subject of the book is developed from scratch except in a few places (e.g. quasiconformal maps) where there exist other very goodbooks: in such cases Pommerenke's emphasis is on giving additional information. There are over two hundred exercises most of which are easy and meant to test the reader's understanding of the text. Each chapter begins with an overview stating the main results informally
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