# Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable

## Lars Ahlfors

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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.

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L'autore:

Retired

Contenuti:

Chapter 1: Complex Numbers

1 The Algebra of Complex Numbers

1.1Arithmetic Operations

1.2Square Roots

1.3Justification

1.4Conjugation, Absolute Value

1.5Inequalities

2 The Geometric Representation of Complex Numbers

2.2The Binomial Equation

2.3Analytic Geometry

2.4The Spherical Representation

Chapter 2: Complex Functions

1 Introduction to the Concept of Analytic Function

1.1Limits and Continuity

1.2Analytic Functions

1.3Polynomials

1.4Rational Functions

2 Elementary Theory of Power Series

2.1Sequences

2.2Series

2.3Uniform Coverages

2.4Power Series

2.5 Abel's Limit Theorem

3 The Exponential and Trigonometric Functions

3.1 The Exponential

3.2 The Trigonometric Functions

3.3 The Periodicity

3.4 The Logarithm

Chapter 3: Analytic Functions as Mappings

1 Elementary Point Set Topology

1.1Sets and Elements

1.2Metric Spaces

1.3Connectedness

1.4Compactness

1.5 Continuous Functions

1.6 Topological Spaces

2 Conformality

2.1Arcs and Closed Curves

2.2Analytic Functions in Regions

2.3Conformal Mapping

2.4Length and Area

3 Linear Transformations

3.1 The Linear Group

3.2 The Cross Ratio

3.3 Symmetry

3.4 Oriented Circles

3.5 Families of Circles

4 Elementary Conformal Mappings

4.1 The Use of Level Curves

4.2 A Survey of Elementary Mappings

4.3 Elementary Riemann Surfaces

Chapter 4: Complex Integration

1 Fundamental Theorems

1.1Line Integrals

1.2Rectifiable Arcs

1.3Line Integrals as Functions of Arcs

1.4Cauchy's Theorem for a Rectangle

1.5 Cauchy's Theorem in a Disk

2 Cauchy's Integral Formula

2.1The Index of a Point with Respect to a Closed Curve

2.2The Integral Formula

2.3Higher Derivatives

3 Local Properties of Analytical Functions

3.1 Removable Singularities. Taylor's Theorem

3.2 Zeros and Poles

3.3 The Local Mapping

3.4 The Maximum Principle

4 The General Form of Cauchy's Theorem

4.1 Chains and Cycles

4.2 Simple Connectivity

4.3 Homology

4.4 The General Statement of Cauchy's Theorem

4.5 Proof of Cauchy's Theorem

4.6 Locally Exact Differentials

4.7 Multiply Connected Regions

5 The Calculus of Residues

5.1 The Residue Theorem

5.2 The Argument Principle

5.3 Evaluation of Definite Integrals

6 Harmonic Functions

6.1 Definition and Basic Properties

6.2 The Mean-value Property

6.3 Poisson's Formula

6.4 Schwarz's Theorem

6.5 The Reflection Principle

Chapter 5: Series and Product Developments

1 Power Series Expansions

1.1Wierstrass's Theorem

1.2The Taylor Series

1.3The Laurent Series

2 Partial Fractions and Factorization

2.1Partial Fractions

2.2Infinite Products

2.3Canonical Products

2.4 The Gamma Function

2.5 Stirling's Formula

3 Entire Functions

3.1 Jensen's Formula

4 The Riemann Zeta Function

4.1 The Product Development

4.2 Extension of ¿(s) to the Whole Plane

4.3 The Functional Equation

4.4 The Zeros of the Zeta Function

5 Normal Families

5.1 Equicontinuity

5.2 Normality and Compactness

5.3 Arzela's Theorem

5.4 Families of Analytic Functions

5.5 The Classical Definition

Chapter 6: Conformal Mapping, Dirichlet's Problem

1 The Riemann Mapping Theorem

1.1Statement and Proof

1.2Boundary Behavior

1.3Use of the Reflection Principle

1.4 Analytic Arcs

2 Conformal Mapping of Polygons

2.1The Behavior at an Angle

2.2The Schwarz-Christoffel Formula

2.3Mapping on a Rectangle

2.4 The Triangle Functions of Schwarz

3 A Closer Look at Harmonic Functions

3.1 Functions with Mean-value Property

3.2 Harnack's Principle

4 The Dirichlet Problem

4.1 Subharmonic Functions

4.2 Solution of Dirichlet's Problem

5 Canonical Mappings of Multiply Connected Regions

5.1 Harmonic Measures

5.2 Green's Function

5.3 Parallel Slit Regions

Chapter 7: Elliptic Functions

1 Simply Periodic Functions

1.1Representation by Exponentials

1.2The Fourier Development

1.3Functions of Finite Order

2 Doubly Periodic Functions

2.1The Period Module

2.2Unimodular Transformations

2.3The Canonical Basis

2.4 General Properties of Elliptic Functions

3 The Weierstrass Theory

3.1 The Weierstrass p-function

3.2 The Functions ¿(z) and s(z)

3.3 The Differential Equation

3.4 The Modular Function ¿(r)

3.5 The Conformal Mapping by ¿(r)

Chapter 8: Global Analytic Functions

1 Analytic Continuation

1.1The Weierstrass Theory

1.2Germs and Sheaves

1.3Sections and Riemann Surfaces

1.4 Analytic Continuations along Arcs

1.5 Homotopic Curves

1.6 The Monodromy Theorem

1.7 Branch Points

2 Algebraic Functions

2.1The Resultant of Two Polynomials

2.2Definition and Properties of Algebraic Functions

2.3 Behavior at the Critical Points

3 Picard's Theorem

3.1 Lacunary Values

4 Linear Differential Equations

4.1 Ordinary Points

4.2 Regular Singular Points

4.3 Solutions at Infinity

4.4 The Hypergeometric Differential Equation

4.5 Riemann's Point of View

Index

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