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Lecture notes on complex analysis. (English) Zbl 1113.30002

Hackensack, NJ: World Scientific (ISBN 1-86094-642-9/hbk; 1-86094-643-7/pbk). xii, 245 p. (2006).
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. It is divided into sixteen chapters, as follows.
The first two chapters deal with the notions of complex numbers and sequences and series, while in the third chapter metric space properties of the complex plane are treated in details. The chapter four is devoted to the theory of analytic functions, while chapter five is concerned with the complex exponential and trigonometric functions, which are defined via their power series expansions. The sixth chapter presents the complex logarithm. In the next two chapters the author presents very rigorously the complex integration and the Cauchy theorem. Chapters nine and ten deal with the Laurent expansion and singularities of meromorphic functions, while in the eleventh chapter the theory of residues is carefully treated. The next chapter is concerned with several problems about zeros and poles of meromorphic functions, the argument principle, Rouché’s theorem and the open mapping theorem. In the chapter 13 there are presented the maximum modulus principle, the Hadamard theorem and the three lines lemma. Chapter 14 deals with Möbius transformations, while the chapter 15 is concerned with the notion of harmonic function. There are included the maximum and minimum principle as well as the local existence of a harmonic conjugate. The last chapter contains results related to local properties of analytic functions. The book ends with an useful appendix, concerning basic results from real analysis that have been needed, and a bibliography.
This book is an excellent textbook, very well written, and enjoyable. It is warmly recommended to students of mathematics and to all researchers that are interested in classical and modern problems in complex analysis.

MSC:

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