Complex Analysis (AMS Chelsea Publishing)

This book is a reprint of the third edition of the classic book on complex analysis. It is a rigorous introduction on an elementary level to the theory of analytic functions of one complex variable and is intended to be used by first year graduate students and advanced undergraduate students. The book covers standard topics in an introductory complex analysis course. The presentation is slanted toward the geometric approach to complex analysis, with a lot of material on conformal mappings, the Riemann mapping theorem, Dirichlet's problem (the existence of a harmonic function with given boundary values), the monodromy theorem, and consideration of the kinds of regions that the Cauchy integral theorem holds for. It also covers such analytic topics as power series, contour integrals, and infinite products. The coverage of special functions is concise but reasonably complete. The presentation is concise, clear, and thorough, and is still fresh today, more than thirty years after its last revision.

Lars Ahlfors