This textbook on complex variables is aimed at the undergraduate population. It covers the topics for such a course completely. The authors include a sufficient number of exercises and questions to help students understand the topic better. However, it is written in a very drab style and many parts feel disconnected, especially in the introductory chapters. The authors fail to capture the reader's interest despite the fascinating topic they cover. The text will serve well as a supplement to a course but should not be the main reading material for the student.
The authors cover the main material for a standard two-semester complex variables course in thirteen chapters. In spite of the title the number of applications to real world problems is minimal.
While mathematically complete, this book lack cohesion; the links between topics that follow each other are often left for the reader to establish. The writing is technical and lacks many descriptive and connective elements. This gives the impression of a collection of facts rather than a whole work.
A good point of the book is the selections of questions and exercises at the end of each section. They are very carefully selected and will help students with reading comprehension and with application of the material of the current section. Most of the time the exercises progress nicely from simple applications of concepts to problems that will expand the student's horizons.
Kai Brunkalla is an Assistant Professor at Walsh University in North Canton, OH.
Preface.- Algebraic and Geometric Preliminaries.- Topological and Analytic Preliminaries.- Bilinear Transformations and Mappings.- Elementary Functions.- Analytic Functions.- Power Series.- Complex Integration and Cauchy's Theorem.- Applications of Cauchy's Theorem.- Laurent Series and the Residue Theorem.- Harmonic Functions.- Conformal Mapping and the Riemann Mapping Theorem.- Entire and Meromorphic Functions.- Analytic Continuation.- Applications.- References.- Index of Special Notations.- Hints for Selected Questions and Exercises.- Index.