Preface

Translator's note

Introduction

1  The geometry of the complex plane

  1.1  The complex plane

  1.2  The geometry of the complex plane

  1.3  The bilinear transformation (M6bius transformation)

  1.4  Groups and subgroups

  1.5  The Riemann sphere

  1.6  The cross-ratio

  1.7  Corresponding circles

  1.8  Pencils of circles

  1.9  Bundles of circles

  1.10  Hermitian matrices

  1.11  Types of transformations

  1.12  The general linear group

  1.13  The fundamental theorem of projective geometry

2  Non-Euclidean geometry

  2.1  Euclidean geometry (parabolic geometry)

  2.2  Spherical geometry (elliptic geometry)

  2.3  Some properties of elliptic geometry

  2.4  Hyperbolic geometry (Lobachevskian geometry)

  2.5  Distance

  2.6  Triangles

  2.7  Axiom of parallels

  2.8  Types of non-Euclidean motions

3  Definitions and examples of analytic and harmonic functions

  3.1  Complex functions

  3.2  Conformal transformations

  3.3  Cauchy-Riemann equations

  3.4  Analytic functions

  3.5  The power function

  3.6  The Joukowsky transform

  3.7  The logarithm function

  3.8  The trigonometric functions

  3.9  The general power function

  3.10  The fundamental theorem of conformal transformation

4  Harmonic functions

  4.1  A mean-value theorem

  4.2  Poisson's integral formula

  4.3  Singular integrals

  4.4  Dirichlet problem

  4.5  Dirichlet problem on the upper half-plane

  4.6  Expansions of harmonic functions

  4.7  Neumann problem

  4.8  Maximum and minimum value theorems

  4.9  Sequences of harmonic functions

  4.10  Schwarz's lemma

  4.11  Liouville's theorem

  4.12  Uniqueness of conformal transformations

  4.13  Endomorphisms

  4.14  Dirichlet problem in a simply connected region

  4.15  Cauchy's integral formula in a simply connected region

5  Point set theory and preparations for topology

  5.1  Convergence

  5.2  Compact sets

  5.3  The Cantor-Hilbert diagonal method

  5.4  Types of point sets

  5.5  Mappings or transformations

  5.6  Uniform continuity

  5.7  Topological mappings

  5.8  Curves

  5.9  Connectedness

  5.10  Special examples of Jordan's theorem

  5.11  The connectivity index

6  Analytic functions

  6.1  Definition of an analytic function

  6.2  Certain geometric concepts

……