Boundary Behaviour of Conformal Maps
Boundary Behaviour of Conformal Maps
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DOI：
10.1007/978-3-662-02770-7
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发表时间：
1992-08
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影响因子：
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通讯作者：
C. Pommerenke
中科院分区：
文献类型：
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作者：
C. Pommerenke
我们研究单位圆盘的共形映射到任意简单连接的平面域上的边界行为。该理论的主要目的是获得函数的解析性质和域的几何性质之间的一一对应关系。在共形映射的经典应用中，域由分段平滑曲线界定。然而，在许多最近的应用程序中，域的边界非常糟糕。它可能没有任何切线，就像 Julia 集的情况一样。那么共形贴图具有许多意想不到的属性，例如几乎所有边界都映射到几乎没有任何东西，反之亦然。本书面向两类用户。(1) 研究生和其他想要了解等形映射的不同级别的人。大多数部分都包含测试理解的练习。它们往往相当简单，只有少数包含新材料。先决条件是一般实数和复数分析，包括有关共形映射的基本事实（例如 AhI66a）。(2) 希望了解共形映射特定方面的非专家，以便找到对其工作有用的东西。因此，大多数章节都以概述开始，概述了一些关键结果，避免了技术问题。这本书并不是对等形映射的详尽概述。必须省略几个重要方面，例如数值方法（参见例如
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 (eg 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, eg numerical methods (see eg