A modern Introduction to Complex Analysis

Question

Is the book Complex variables by Carlos A Berenstein and Roger Gay a good book for a second, more rigorous, course in complex analysis? My first course, while I loved the applications to analytic number theory, felt dry and bland in its analysis as they just gave definitions. This book seems to be one of its kind as it talks about algebraic topology, homological algebra, and sheaf theory while developing complex analysis from the definition of a holomorphic funtion (It gives a really good reason from linear algebra why we have the definition of a holomorphic function the way we do). I can't find any reviews or really anything about this book except for one reply on the page. Is this a good book to read or is there a similar one that covers a modern approach like it? (By the way my first course text was Complex analysis by Elias M. Stein which I read up to chapter 7).

Answer 1

I think this is a challenging, highly readable book, worth a detailed study.

In order to get an impression about the contents of the book, the authors write in the preface:

• ... not only complex analysis in several variables, but also number theory, harmonic analysis, and other branches of mathematics, both pure and applied, have required a reconsideration of analytic continuation, ordinary differential equations in the complex domain, asymptotic analysis, iteration of holomorphic functions, and many other subjects from the classic theory of functions of one complex variable.

  This ongoing reconsideration led us to think that a textbook incorporating some of these new perspectives and techniques had to be written. In particular, we felt that introducing ideas from homological algebra, algebraic topology, sheaf theory, and the theory of distributions, together with the systematic use of the Cauchy-Riemann $\overline{\partial}$-operator, were essential to a complete understanding of the properties and applications of the holomorphic functions of one variable.

Prerequisites, the reader should know:

• ... We have tried to make this book self-contained and accessible to graduate students, while at the same time to reach quite far into the topics considered.

  For that reason we assume mainly knowledge that is found in the undergraduate curriculum, such as elementary linear algebra, calculus, and point set topology for the complex plane and the the two-dimensional sphere $S^2$.

  Beyond this, we assume familiarity with metric spaces , the Hahn-Banach theorem, and the theory of integration as it can be found in many introductory texts of real analysis. Whenever we felt a subject was not universally known, we have given a short review of it.

Note: A review by Nick Lord has been published in The Mathematical Gazette Volume 78 Issue 482.