The Introduction to Complex Analysis by Tutschke and Vasudeva is anything but an “introduction”. It is in fact one of the most comprehensive books on complex analysis that I have come across. The book packs a dense 453 pages in which it covers all the standard topics of one variable complex analysis, ending with a chapter on special functions and one on boundary value problems.
The writing is crisp, correct, and very well organized. Sometimes alternate approaches are presented, the most notable being the development of the complex differentiation and integration, ending with Cauchy’s Integral Formula. This is done first in chapter two in a classical manner, based on the Cauchy-Riemann system written in terms of the real and imaginary parts of a complex function, and then again in chapter three, based on complex partial derivatives. Throughout the book the approach is analytic/algebraic. There are few figures, and virtually no geometric interpretations. Combined with the relatively few exercises (151 problems distributed throughout the twelve chapters of the book), this book does not make for an easy read, and I would not recommend it as a first textbook, even for a graduate level course.
However, the book would make a great reference for anybody who has previous knowledge of complex analysis. The examples are well chosen to prove the points of the statements they are accompanying. Also, the encyclopedic nature of the book guarantees that the reader will find all the results and proofs of classical complex analysis included. For example, the reader will find a discussion and proof of the Riemann mapping theorem, a proof of a special case of the Bieberbach conjecture, detailed discussions of elliptic functions, and constructions of solutions for certain types of Dirichlet boundary value problems.
Overall, Introduction to Complex Analysis is a solid, well-written reference book for classical complex analysis.
Ioana Mihaila (email@example.com) is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is also interested in mathematics competitions.