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.

1.The Algebra of Complex Numbers

2.The Geometric Representation of Complex Numbers

CHAPTER2.COMPLEX FUNCTIONS

1.Introduction to the Concept of Analytic Function

2.Elementary Theory of Power Series

3.The Expononential and Trigonometric Functions

CHAPTER3.ANALYTIC FUNCTIONS AS MAPPINGS

1.Elementaty Point Set Topology

2.Conformality

3.Linear Transformations

4.Elementary Conformal Mappings 

CHAPTER4.COMPLEX INTEGRATION

1.Fundamental Theorems

2.Cauchy's Integral Formula

3.Local Properies of Analytical Functions

4.The General Form of Cauchy's Theorem

5.The Calculus of Residues

6.Harmonic Functions

CHAPTER 5.SERIES AND PRODUCT DEVELOPNEMTS

1.Power Series Expansions

2.Partial Fractions and Factorization

3.Entire Functions

4.The Riemann Zeta Function

5.Normal Families

CHAPTER 6.CONFORMAL MAPPING.DIRICHLET'S PROBLEM

1.The Riemann Mapping Theorem 

2.Conformal Mapping of Polygons

3.A Closer Look at Harmonic Functions

4.The Dirichlet Problem

5.Canonical Mappings of Multiply Connected Regions

CHAPTER 7. ELLIPTIC FUNCTIONS

1.Simply Periodic Functions

2.Doubly Periodic Functions

3.The Weierstrass Theory 

CHAPTER 8.GLOBAL ANALTIC FUNCTIONS

1.Analytic Continuation

2.Algebraic Functions

3.Picard's Theorem

4.Linear Differential Equations

Lars V. Ahlfors

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