This book is intended as a textbook for a two-semester course in complex analysis aimed at beginning graduate students or advanced undergraduates with a background in real analysis. The emphasis is on the proofs of the main results: local and global Cauchy theory and its applications, the Riemann mapping theorem, and the Weierstrass factorization theorem. The book is suitable as a textbook and also for the independent reading, since it contains complete explanations, many nice problems, and full solutions.
This second edition of Ash's 1971 book keeps the clear, crisp, and concise style of the first edition while introducing several improvements: a clear explanation of the relationship between real-differentiability and the Cauchy-Riemann equations, which is not always explicit in many texts; Dixon's proof of the homology version of Cauchy's theorem, with full details; the use of hexagonal tilings to characterize simple connectedness in terms of winding numbers; Grabiner's proof of Runge's theorem; and Korevaar's version of Newman's proof of the Prime Number Theorem.
The book is also available in PDF at Ash's web site, but reading an inexpensive good-quality hard copy such as Dover's is much better than printing your own. (One minor issue is that the index and the list of symbols in the Dover edition still use the PDF page numbers, which start from 1 at each chapter.)
Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.