Includes index.

Chapter 1: Complex Numbers1 The Algebra of Complex Numbers1.1 Arithmetic Operations1.2 Square Roots1.3 Justification1.4 Conjugation, Absolute Value1.5 Inequalities2 The Geometric Representation of Complex Numbers2.1 Geometric Addition and Multiplication2.2 The Binomial Equation2.3 Analytic Geometry2.4 The Spherical RepresentationChapter 2: Complex Functions1 Introduction to the Concept of Analytic Function1.1 Limits and Continuity1.2 Analytic Functions1.3 Polynomials1.4 Rational Functions2 Elementary Theory of Power Series2.1 Sequences2.2 Series2.3 Uniform Coverages2.4 Power Series2.5 Abel's Limit Theorem3 The Exponential and Trigonometric Functions3.1 The Exponential3.2 The Trigonometric Functions3.3 The Periodicity3.4 The LogarithmChapter 3: Analytic Functions as Mappings1 Elementary Point Set Topology1.1 Sets and Elements1.2 Metric Spaces1.3 Connectedness1.4 Compactness1.5 Continuous Functions1.6 Topological Spaces2 Conformality2.1 Arcs and Closed Curves2.2 Analytic Functions in Regions2.3 Conformal Mapping2.4 Length and Area3 Linear Transformations3.1 The Linear Group3.2 The Cross Ratio3.3 Symmetry3.4 Oriented Circles3.5 Families of Circles4 Elementary Conformal Mappings4.1 The Use of Level Curves4.2 A Survey of Elementary Mappings4.3 Elementary Riemann SurfacesChapter 4: Complex Integration1 Fundamental Theorems1.1 Line Integrals1.2 Rectifiable Arcs1.3 Line Integrals as Functions of Arcs1.4 Cauchy's Theorem for a Rectangle1.5 Cauchy's Theorem in a Disk2 Cauchy's Integral Formula2.1 The Index of a Point with Respect to a Closed Curve2.2 The Integral Formula2.3 Higher Derivatives3 Local Properties of Analytical Functions3.1 Removable Singularities. Taylor's Theorem3.2 Zeros and Poles3.3 The Local Mapping3.4 The Maximum Principle4 The General Form of Cauchy's Theorem4.1 Chains and Cycles4.2 Simple Connectivity4.3 Homology4.4 The General Statement of Cauchy's Theorem4.5 Proof of Cauchy's Theorem4.6 Locally Exact Differentials4.7 Multiply Connected Reg

A standard source of information of functions of one complex variable, this text has retained its wide popularity in this field by being consistently rigorous without becoming needlessly concerned with advanced or overspecialized material. Difficult points have been clarified, the book has been reviewed for accuracy, and notations and terminology have been modernized. Chapter 2, Complex Functions, features a brief section on the change of length and area under conformal mapping, and much of Chapter 8, Global-Analytic Functions, has been rewritten in order to introduce readers to the terminology of germs and sheaves while still emphasizing that classical concepts are the backbone of the theory. Chapter 4, Complex Integration, now includes a new and simpler proof of the general form of Cauchy's theorem. There is a short section on the Riemann zeta function, showing the use of residues in a more exciting situation than in the computation of definite integrals.