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."
General
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