Complex Variables: Second Edition (Dover Books on Mathematics) by Robert B. Ash (PDF)

Description

This text on complex variables is geared toward graduate students and undergraduates who have taken an introductory course in real analysis. It is a substantially revised and updated edition of the popular text by Robert B. Ash, offering a concise treatment that provides careful and complete explanations as well as numerous problems and solutions.An introduction presents basic definitions, covering topology of the plane, analytic functions, real-differentiability and the Cauchy-Riemann equations, and exponential and harmonic functions. Succeeding chapters examine the elementary theory and the general Cauchy theorem and its applications, including singularities, residue theory, the open mapping theorem for analytic functions, linear fractional transformations, conformal mapping, and analytic mappings of one disk to another. The Riemann mapping theorem receives a thorough treatment, along with factorization of analytic functions. As an application of many of the ideas and results appearing in earlier chapters, the text ends with a proof of the prime number theorem.

User’s Reviews

Editorial Reviews: About the Author Robert B. Ash is Professor Emeritus of Mathematics at the University of Illinois. W. P. Novinger, now retired, was Professor of Mathematics at Florida State University.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐good but, dense & intenseGood to have other books around for reference. Like RBA (Robert B. Ash) Real Variables, RAS (Richard A. Silverman) Complex Analysis w/ Applications (gentle). Then, when comprehended RBA RV & CV, and RAS CAw/A, you’re ready for John W. Dettman (JWD), Applied Complex Variables. All published by Dover. For about $45 you get a course on Complex Variables.

⭐My search for a complex analysis text that I could work through on my own from front to back finally ended with this gem. I had first tried Conway’s “Functions of One Complex Variable,” but kind of sputtered out around 80 pages in. I then was subjected to the nightmarish wad of worthless pulp written by Ahlfors (which I reviewed) — thank god a semester lasts only 15 weeks. Then I tried Lang’s “Complex Analysis” text, and again only got about 80 pages in. Lang’s treatment of formal power series lacked a firm theoretical underpinning, and so the whole edifice he was trying to construct ultimately sat on a hill of sand. Someday I’ll review Lang’s book, maybe. Finally I found Ash & Novinger.This book is tight. It holds together logically, from one section to the next. It certainly assumes thorough knowledge of analysis up to the level of the first eight chapters of Rudin’s “Principles of Mathematical Analysis” [Rud], but that is reasonable. A proof won’t mention that it’s using the Cauchy Criterion for uniform convergence of a series of functions, which is in [Rud], p. 147. Typos are few, and none are serious. In the back of the book there are solutions, or at least hints, to all the exercises.A few critical things are undefined. A precise definition of a Laurent series, and what it means for a Laurent series to converge, is missing. This is a common oversight in many complex analysis books, for some reason. Also a clear definition of what it means to be analytic at infinity is somewhat lacking. There is a definition, but it’s not stated quite as nicely as: “A function f is analytic at infinity if it has a removable singularity at infinity.” Why can’t authors of complex analysis texts just say this? It’s not hard. In addition, there is lacking a definition for what it means for a series of functions to be “absolutely convergent,” something not discussed in [Rud]. There are other small information gaps, but other books or the Internet always filled in those gaps. Subtract a quarter star for this.Subtract another quarter star for a general lack of exercises in the last three chapters, and also some slightly unclear exposition in the sections on the extended complex plane (section 3.4) and families of functions (chapter 5). Both issues I was able to resolve on my own, but I’m not sure if I could have done it if I didn’t have extensive experience with the finer points of point-set topology and metric spaces.I’d love to see this book go into a 3rd addition with some expanded exposition and additional exercises. It would also be great to see Ash & Novinger add some additional topics — say a chapter on the Picard theorems and such. To think, you could buy 32 copies of this book with the money that would buy just one copy of Ahlfors at its current price of $240!

⭐Modern, broad, and entertaining exposition of the classical parts of the theory of analytic functions. I like everything that Professor Robert B. Ash wrote. His textbook “Real Analysis and Probability,” for example, is a gem (I prefer the first edition).This is an ideal textbook for a semester long upper division undergraduate or beginning graduate course. The pitch is that of a pure math course, not much of an applied one (for this see the text by Ablowitz & Fokas). Although the book is freely available on the internet, I’d buy (and, in fact, did buy) the Dover edition at great price. A no-brainer.

⭐Think of this text as a workbook. There are more encyclopedic references for this material, but Ash has gifted this volume to students of mathematics; and this is absolutely beautiful mathematics. Heartily recommended for those who wish to learn the theory.

⭐This is a nice, rather terse introduction to complex analysis. The style of the book is rather modern, incorporating many of the more recent insights and exposition.

⭐This Dover edition is an updated, improved and more comprehensive version of a former work by the first author, which was indeed a very good presentation of the most important topics in the theory of one variable holomorphic functions. The Dirichlet’s problem is dealt with using Poisson’s formula. Jensen’s formula on the zeroes of an holomorphic function on a disk is given. Integer functions and infinite products are introduced, although not fully covered (no Hadamard’s theorem). Meromorphic functions are introduced too, and Mittag-Leffler’s theorem is proved. However, elliptic functions are not dealt with here, what is a pity. But the merits of the book are immense: in only a few pages the authors provide a very accurate and elegant exposition of the main topics, (preferring the Cauchy integration method to Weierstrass power series approach, almost everywhere). Riemannn’s theorem proof is extremely short. Caratheodory technique of extending conformal maps to the boundary is explained. Finally the prime number theorem is proved, following the method of D. J. Newmann, and giving us the opportunity to admire Riemann’s zeta function at work. Solutions of problems are added at the end. This book is a bargain and a jewel. And it has only two hundred pages! Amazing. Maybe, power series and residues deserved a bit more attention, but as it is, this book is close to perfection.

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