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Selected Topics in the Classical Theory of Functions of a Complex Variable

Maurice Heins
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mehdi Hassani
, on

The discovery and development of the theory of functions of a complex variable proceeded in parallel to other progress in mathematics. Indeed, a remarkable part of that progress was the discovery of deep connections between complex analysis and other branches of mathematics. An important example is the realization that the so-called Riemann zeta function should be considered as a function of a complex variable, which finally led to a proof of the prime number theorem and opened up several paths to new mathematics, many of which are still being pursued.

The central role of the theory of functions of a complex variable in mathematics has motivated many authors to write books on the subject. Most of them contain some famous results, but usually follow a similar structure which of necessity leaves out a large number of important results. The book under review presents a selection of topics in the theory of complex functions of a single variable, including some of the results that are often omitted from the standard texts.

The book assumes a basic knowledge of the theory, moving quickly to specialized topics. It consists of short sections that focus on a specific theorem and give some related results, usually as exercises. Almost all the exercises are proofs rather than computations, which allows the book to be very concise.

The topics are arranged in six chapters. Chapter 1 reviews some preliminaries. Chapter 2 studies some properties of meromorphic functions. Chapter 3 is on Picard’s Theorem and some related results. Chapter 4 considers harmonic functions. In this chapter, the author discusses properties of harmonic functions, the Poisson-Stieltjes integral and Fatou’s Theorem, subharmonic functions and their applications, and the Dirichlet problem. In Chapter 5, the author considers several mathematical applications, and in Chapter 6, he studies the boundary behavior of the Riemann mapping function for simply-connected Jordan regions.

This book seems very useful for people searching for classical results in the theory of functions of a complex variable that go beyond the standard introductory texts. 

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.

The table of contents is not available.