Function theory of one complex variable. 3rd ed.

*(English)*Zbl 1114.30001
Graduate Studies in Mathematics 40. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3962-4/hbk). xix, 504 p. (2006).

The book whose first edition was published in 1997 is very well written, it is extremely clear and very detailed. It is suitable to be used as a textbook in a first-year graduate course in complex analysis, as it covers all material usually treated in such a course. In addition to this, the book includes a number of special topics such as Hardy spaces, the Bergman kernel function and the Bell-Ligocka approach to proving smoothness to the boundary of biholomorphic mappings.

One of the features of the book is that it emphasizes the connection between real and complex analysis. The authors assert in the preface to the first edition that the easiest path into complex analysis is to observe how at least its rudiments arise directly from ideas about calculus which with the student must be familiar. Consequently, they pursue this point of view both by comparing and by constrasting complex variable theory with real-variable claculus.

We should also mention that the distinct chapters or sections include very nice introductions which illuminate the subjects involved from several sides.

The book contains many solved examples and a good number of exercises as an essential part of it. In this edition, many of the exercises have been revised and in some cases rearranged for consistency. Also a considerable number of the proofs, especially in the later chapters, have been corrected, clarified, or simplified.

I can say that I have read this book with great pleasure and I do recommend it for those who are interested in complex analysis.

One of the features of the book is that it emphasizes the connection between real and complex analysis. The authors assert in the preface to the first edition that the easiest path into complex analysis is to observe how at least its rudiments arise directly from ideas about calculus which with the student must be familiar. Consequently, they pursue this point of view both by comparing and by constrasting complex variable theory with real-variable claculus.

We should also mention that the distinct chapters or sections include very nice introductions which illuminate the subjects involved from several sides.

The book contains many solved examples and a good number of exercises as an essential part of it. In this edition, many of the exercises have been revised and in some cases rearranged for consistency. Also a considerable number of the proofs, especially in the later chapters, have been corrected, clarified, or simplified.

I can say that I have read this book with great pleasure and I do recommend it for those who are interested in complex analysis.

Reviewer: Daniel Girela (MĂˇlaga)

##### MSC:

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |