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Title: Complex Variables and Applications
ISBN: 0072872527
Author:
James Ward Brown
Ruel Vance Churchill
Publicate Date: 2003-02-26
Publish: 2003-02-26
List Price: $39.75
Average Customer Rating: 4.0
Format: Hardcover
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Amazon Lowest New Price: $39.75
Amazon Lowest Used Price: $20.00
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| Customer Review: |
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1: Merely adequate.
I purchased this book because the undergraduate course I took in complex analysis was taught by a professor who preferred to use Schaum's Outlines: Complex Variables (With an Introduction to Conformal Mapping and Its Applications) accompanied by some fabulous lectures. I didn't save my lecture notes, though, and I wanted a more thorough refresher in the subject than Schaum's can give. So, my qualifications when turning to this text are the following: undergraduate degree in math, previous experience with complex analysis, more extensive experience with real analysis, very recent review of multivariable calculus (which I mention because of the numerous parallels between some of the line integral theorems and contour integrals in the complex plane).
When I first picked up the book, it wasn't quite what I hoped for. Very short sections are divided into well-organized chapters. The sections themselves are hit-or-miss in terms of both depth and breadth of material. Some sections deal with a topic that seems meaty enough to warrant its own treatment (branch cuts and branch points) but without going into anything near the detail necessary to use the concept; others devote an entire section to a single theorem (Cauchy-Goursat) and another section to its proof; others combine several new ideas in one section devoted to treating a larger concept, the way most mathematics texts do. These sections are, unfortunately, few and far-between. In skimming superficially over an important topic or ponderously plodding through a single theorem without tying it to other material, the authors have created a book that feels disorganized and nebulous.
For my purposes--review of a subject with which I am already passingly familiar--this text works fine. I can see connections before they're introduced because I already know where the theory is headed, and my previous experience with mathematics makes it easier to see how things fit into place in the larger framework of analysis. But I have to wonder how an undergraduate with no previous experience in complex analysis would fare using this text. Concepts are introduced before they're used, and some material that I thought was pretty complex (pardon the pun, har har) is glossed over as if it were completely obvious.
The poor organization and layout contributes to the difficulty in comprehension. While the chapters are well-defined, and the sections are at least labeled by topic and numbered, I don't see how you could find your way through this text without copious highlighting. Theorems are offset with a nice bold "theorem/corollary/lemma" in front of them, but several are typeset to take up several lines so that, after the first paragraph break, it's easy to miss where the theorem ends and the discussion begins. The proofs are even worse. I never thought I'd yearn so desperately for three simple letters, but QEDs are completely missing from this text. Proofs go on for paragraphs, often interrupted by figures or even examples, without any sign from the layout that a conclusion has been reached or a new topic begun. Figures are often useful but poorly placed, so that the material referencing them is on a totally different portion of the page. Some theorems are stated more conversationally than elegantly, but at least that means I get to practice rephrasing in my notes.
The exercises in this text are very helpful. Examples are interspersed with the theory, often providing immediate applications and almost always assisting with the exercises at the end of the section. The exercises themselves are quite frequently guided with hints as to how to proceed (particularly with proofs) or accompanied by answers to enable work-checking. The progression of exercises is also very natural, working from simpler concepts to more advanced ones in a way that doesn't overwhelm the student.
Generally, I would recommend this book to someone hoping to review a subject that they already have some understanding of. Enough of the theory is obtuse enough that I wouldn't recommend it to someone who was looking for something to help them better understand the subject, but, unfortunately, I also can't think of a BETTER text. Overall, this book has no killing flaws. It does what it sets out to do. I just can't imagine how the eighth edition manages to have organizational flaws and skimpy detail after seven previous editions for students to complain about.
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2: Useful, but I hate it.
I really hated this book enough to give it 1 star, but I have learned so much from it. It is true that complex analysis is harder than real analysis, and requires extra care in learning it, but I have always found it difficult to enjoy what I am reading.
I have read 3/4 of the book, tried to reproduce most of the proofs, and solve all of the excercises in the sections that I read for a Phd qualifying exam. Most of my colleagues recommend Asmar's Applied Complex Analysis with Partial Differential Equations to this one. I have tried to take a look at it occassionaly. It looked Ok, but I can't be a good judge of that.
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3: Very poor book
Brown and Churchill's book is neither rigorous nor intuitive; it is a true pedagogical nightmare. The authors are extremely sloppy with their exposition, structure and rigor.
Some trivial results are "proved" with pedantic detail, but even there the proof is not exhaustive. As an example, to prove sufficient conditions for differentiability (p. 63-5), two pages are devoted to setting up some elaborate structure, but the real meat (basically Taylor's theorem) is not even mentioned, rather the authors cite another text. Similarly, equality of mixed partial derivatives is waved off as "a theorem in advanced calculus." What is complex analysis but advanced calculus, and why do the authors here devote space to prove thoroughly trivial results (e.g. limit of sums converges to sum of limits), while leaving out other important foundations?
A similar sloppiness is shown even in those results that are more fully proved in the book. For example, the theorem presented in Section 26 depends on a theorem in Section 68! Pedagogically this is inexcusable, as the authors introduce these results willy-nilly, not as a coherent whole; the book must be read at least twice to check its consistency!
The layout of the book is awfully confusing. There is practically no white space, and a single font and font size are used throughout the book, for explanations, theorems, examples and exercises. Examples sometimes are placed within the section they illustrate, and sometimes bizarrely they are given their own section. This means that the table of contents cannot indicate the relative importance of book content. Likewise proofs are sometimes given their own sections, sometimes buried in an overly large section.
The exercises are mostly computational, and usually they are spoiled by "hints" that are so exhaustive that the only thing left for the student to do is to move some symbols around as directed by the book.
In general, the book causes both my mathematically rigorous colleagues and my application oriented colleagues to cringe in pain. Compared to some other works on analysis, this volume is a true abomination. Walter Rudin's Principles of Mathematical Analysis is an exquisite, mathematically thorough and rigorous treatise on the subject, in which practically every exercise is meaningful. That the publisher of the present book dares to charge as much money for this seventy-year old volume as Rudin's book costs is farcical.
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4: It is what it is
God knows this book is used by everybody and their brother to teach Complex Variables. Why, I do not know. Dry as dust and even more boring. "Proofs" are minimal and the exercises are plug and chug. Still, it has the acceptance of academia. Enjoy.
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5: Better for Engineering, Physics, and others in the Applied areas, than for pure mathematics students
This is a nice book for understanding the basic concepts of early Complex Analysis. The integral formulas, residue theorems, Fourier Analysis, infinite series/sequences, etc. are all covered. There are plenty of exercises and examples. Everything is so clearly presented, that it is easy for people with very little background in analysis to read (ie, you don't really need real analysis before reading this book).
My problems with the book are thus: There are very, very few proofs to any of the theorems. I'd rather have more proofs than examples. The problems are almost all computational. Almost none of the exercises require much thought, although some of them will take a while to do. There is no discussion on the importance of certain topics to the wider context of math. No discussion of certain standard complex-valued functions like elliptic functions, zeta functions, or gamma functions.
If anything, I see this as mostly a how-to book for engineers and physicists who come across complex variables in their work. For math students, I'd recommend the books by Shakarchi/Stein, Lang, Conway, Ahlfors.
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