Ebook: Boundary Behaviour of Conformal Maps

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.

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There has been a great deal of recent interest in the boundary behaviour of conformal maps of the unit disk onto plane domains. In classical applications of conformal maps, the boundary tended to be smooth. This is not the case in many modern applications (e.g. for Julia sets). The first chapters present basic material and are also of interest for people who use conformal mapping as a tool. The later chapters deal in greater detail with classical material and, go into recent developments (e.g. by Makarov). The reader is assumed to know standard complex and real analysis. The subject of the book is developed from scratch except in a few places (e.g. quasiconformal maps) where there exist other very goodbooks: in such cases Pommerenke's emphasis is on giving additional information. There are over two hundred exercises most of which are easy and meant to test the reader's understanding of the text. Each chapter begins with an overview stating the main results informally

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There has been a great deal of recent interest in the boundary behaviour of conformal maps of the unit disk onto plane domains. In classical applications of conformal maps, the boundary tended to be smooth. This is not the case in many modern applications (e.g. for Julia sets). The first chapters present basic material and are also of interest for people who use conformal mapping as a tool. The later chapters deal in greater detail with classical material and, go into recent developments (e.g. by Makarov). The reader is assumed to know standard complex and real analysis. The subject of the book is developed from scratch except in a few places (e.g. quasiconformal maps) where there exist other very goodbooks: in such cases Pommerenke's emphasis is on giving additional information. There are over two hundred exercises most of which are easy and meant to test the reader's understanding of the text. Each chapter begins with an overview stating the main results informally
Content:
Front Matter....Pages I-IX
Some Basic Facts....Pages 1-17
Continuity and Prime Ends....Pages 18-40
Smoothness and Corners....Pages 41-70
Distortion....Pages 71-93
Quasidisks....Pages 94-125
Linear Measure....Pages 126-154
Smirnov and Lavrentiev Domains....Pages 155-172
Integral Means....Pages 173-194
Curve Families and Capacity....Pages 195-221
Hausdorff Measure....Pages 222-244
Local Boundary Behaviour....Pages 245-267
Back Matter....Pages 269-300

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There has been a great deal of recent interest in the boundary behaviour of conformal maps of the unit disk onto plane domains. In classical applications of conformal maps, the boundary tended to be smooth. This is not the case in many modern applications (e.g. for Julia sets). The first chapters present basic material and are also of interest for people who use conformal mapping as a tool. The later chapters deal in greater detail with classical material and, go into recent developments (e.g. by Makarov). The reader is assumed to know standard complex and real analysis. The subject of the book is developed from scratch except in a few places (e.g. quasiconformal maps) where there exist other very goodbooks: in such cases Pommerenke's emphasis is on giving additional information. There are over two hundred exercises most of which are easy and meant to test the reader's understanding of the text. Each chapter begins with an overview stating the main results informally
Content:
Front Matter....Pages I-IX
Some Basic Facts....Pages 1-17
Continuity and Prime Ends....Pages 18-40
Smoothness and Corners....Pages 41-70
Distortion....Pages 71-93
Quasidisks....Pages 94-125
Linear Measure....Pages 126-154
Smirnov and Lavrentiev Domains....Pages 155-172
Integral Means....Pages 173-194
Curve Families and Capacity....Pages 195-221
Hausdorff Measure....Pages 222-244
Local Boundary Behaviour....Pages 245-267
Back Matter....Pages 269-300
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