Boundary Behaviour of Conformal Maps

• Christian Pommerenke
Book

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 299)

1. Front Matter
Pages I-IX
2. Christian Pommerenke
Pages 1-17
3. Christian Pommerenke
Pages 18-40
4. Christian Pommerenke
Pages 41-70
5. Christian Pommerenke
Pages 71-93
6. Christian Pommerenke
Pages 94-125
7. Christian Pommerenke
Pages 126-154
8. Christian Pommerenke
Pages 155-172
9. Christian Pommerenke
Pages 173-194
10. Christian Pommerenke
Pages 195-221
11. Christian Pommerenke
Pages 222-244
12. Christian Pommerenke
Pages 245-267
13. Back Matter
Pages 269-300

Introduction

We study the boundary behaviour of a conformal map of the unit disk onto an arbitrary simply connected plane domain. A principal aim of the theory is to obtain a one-to-one correspondence between analytic properties of the function and geometrie properties of the domain. In the classical applications of conformal mapping, the domain is bounded by a piecewise smooth curve. In many recent applications however, the domain has a very bad boundary. It may have nowhere a tangent as is the case for Julia sets. Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa. The book is meant for two groups of users. (1) Graduate students and others who, at various levels, want to learn about conformal mapping. Most sections contain exercises to test the understand­ ing. They tend to be fairly simple and only a few contain new material. Pre­ requisites are general real and complex analyis including the basic facts about conformal mapping (e.g. AhI66a). (2) Non-experts who want to get an idea of a particular aspect of confor­ mal mapping in order to find something useful for their work. Most chapters therefore begin with an overview that states some key results avoiding tech­ nicalities. The book is not meant as an exhaustive survey of conformal mapping. Several important aspects had to be omitted, e.g. numerical methods (see e.g.

Keywords

Smooth function boundary behaviour complex analysis conformal map differential equation partial differential equations

Authors and affiliations

• Christian Pommerenke
• 1
1. 1.Fachbereich MathematikTechnische UniversitätBerlin 12Germany

Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-02770-7
• Copyright Information Springer-Verlag Berlin Heidelberg 1992
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-08129-3
• Online ISBN 978-3-662-02770-7
• Series Print ISSN 0072-7830