Geometric Function Theory: A Second Course in Complex Analysis

Preface
Contents
1 Introduction
	1.1 Why Study Complex Analysis?
	1.2 Prerequisites
	1.3 Overall Summary of the Contents
	1.4 What Is Not in This Book
	1.5 Notation
2 The Complex Plane: Preparatory Topics
	2.1 Circles and Straight Lines in the Complex Plane
		2.1.1 A Parameterised Family of Circles
	2.2 Möbius Transformations
	2.3 The Schwarz Reflection Principle
		2.3.1 The Schwarz Reflection Principle for Lines
		2.3.2 Reflection in a Circle
		2.3.3 The Schwarz Reflection Principle for Circles
	2.4 Singularities of Analytic Functions
3 The Riemann Sphere
	3.1 Stereographic Projection
		3.1.1 Circles Under Stereographic Projection
	3.2 The Topology of the Extended Complex Plane
	3.3 The Riemann Sphere
		3.3.1 The Riemann Sphere as a Riemann Surface
		3.3.2 Analytic Functions on the Riemann Sphere
		3.3.3 Meromorphic Functions on the Riemann Sphere
	3.4 The Spherical Metric
		3.4.1 General Construction of a Riemannian Metric on a Planar Domain
		3.4.2 The Infinitesimal Spherical Metric
		3.4.3 The Spherical Metric on C∞
		3.4.4 Isometries of the Spherical Metric
		3.4.5 Geodesics for the Spherical Metric
	3.5 Some Spherical Geometry
4 The Hyperbolic Disk
	4.1 The Basic Schwarz Lemma
	4.2 Automorphisms of the Disk
	4.3 Automorphisms of a Half-Plane
	4.4 The Schwarz-Pick Lemma: First Version
	4.5 The Hyperbolic Metric in the Disk
		4.5.1 Construction of the Hyperbolic Metric
		4.5.2 Geodesics, and a Formula for Hyperbolic Distance
	4.6 The Schwarz-Pick Lemma: Second Version
	4.7 The Hyperbolic Metric in a Half-Plane
5 Normal Families and Value Distribution
	5.1 The Arzelà-Ascoli Theorem
	5.2 The Space of Continuous Functions on a Planar Domain
	5.3 Convergence of Sequences of Analytic and Meromorphic Functions
	5.4 The Theorems of Montel, Marty and Zalcman
	5.5 Montel\'s Second Theorem and the Great Picard Theorem
6 Simply Connected Domains, the Riemann Mapping Theorem and Conformal Mapping
	6.1 The Homotopic Form of Cauchy\'s Theorem
	6.2 Simply Connected Domains
	6.3 The Riemann Mapping Theorem
	6.4 The Hyperbolic Metric in a Simply Connected Domain
		6.4.1 Construction of the Hyperbolic Metric
		6.4.2 The Schwarz-Pick Lemma: Third Version
		6.4.3 Monotonicity of the Hyperbolic Metric
	6.5 Some Special Domains and Their Hyperbolic Metrics
		6.5.1 The Half-Plane
		6.5.2 The Strip
		6.5.3 Sectors, Including the Slit-Plane
		6.5.4 The Half-Disk and the Slit-Disk
	6.6 Boundary Correspondence Under Conformal Mappings
		6.6.1 Proof of Theorem 6.12
	6.7 Conformal Mapping of Annuli
7 Runge\'s Theorem and Further Characterisations of Simply Connected Domains
	7.1 Runge\'s Theorem
	7.2 A General Integral Formula
	7.3 Winding Numbers and Further Characterisations of Simply Connected Domains
8 Univalent Functions: The Basics
	8.1 The Classes S and  of Univalent Functions
	8.2 Bieberbach\'s Coefficient Estimate and the Koebe1/4-Theorem
		8.2.1 The Area Theorem and Bieberbach\'s Coefficient Estimate
		8.2.2 The Koebe 1/4-Theorem
	8.3 Growth and Distortion Theorems for Functions in the Class S
9 Carathéodory Convergence of Domains and Hyperbolic Geodesics
	9.1 Carathéodory Convergence of Domains
	9.2 Comparison Results for the Hyperbolic Metric
		9.2.1 The Laplacian and the Hyperbolic Metric
		9.2.2 Estimates for Solutions of u =e2u
		9.2.3 Half-Planes and Disks Are Hyperbolically Convex
10 Uniformisation of Planar Domains
	10.1 Covering Spaces and the Monodromy Theorem
	10.2 The Modular Function
		10.2.1 A Series of Reflections
		10.2.2 Proof of Theorem 10.5: The Reflected Mappings
		10.2.3 The Automorphism Group of the Modular Function
	10.3 The Picard Theorems: Reprise
	10.4 Uniformisation of Planar Domains
Glossary
Solutions to the Exercises
	Exercises from Chap.2
	Exercises from Chap.3
	Exercises from Chap.4
	Exercises from Chap.5
	Exercises from Chap.6
	Exercises from Chap.7
	Exercises from Chap.8
	Exercises from Chap.9
	Exercises from Chap.10
References
Index