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A definite need exists for an introduction to functions of complex variables in which all results are derived from a simple set of axioms. Since no such book is available I have attempted to fill the gap by writing a text in which occur neither “intuitiveproofs” nor theorems for whose proofs the reader is referred to other sources. The only intentional exception to this rule is the omission of those proofs that seemed to be simple enough to be left as exercises for the reader. In certain borderline cases outlines of proofs are given. Thus the statement that no previous mathematical knowledge is required of the reader is literally true. It is hoped that this book will be useful as a text. I have used preliminary drafts of it in a two-semester course at Washington University in the academic years 1946-47 and 1948-49. It also is intended to serve as a reference book to which instructors and interested students may want to turn for complete derivations in many instances not to be found elsewhere. Fundamental concepts Real numbers Cardinal numbers Complex numbers Sums and products Hausdorff spaces Metric spaces The plane of complex numbers Limits, continuity, differentiability Real functions of real variables Curves and regions in the plane of complex numbers Some combinatorial topology Jordan curves Rectifiable and directed curves Integration The Cauchy integral theorem Sequences of functions Infinite series Power series Function spaces Elementary transcendental functions Inverse functions Analytic continuation and singular points The extended plane Classification of some single-valued analytic functions Residues Conformal maps Linear fractional transformations and inversions Simple functions and the Riemann mapping theorem The elliptic modular function and Picard’s theorems Riemann surfaces Index