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Conformal Mapping of Simply and Multiply Connected Regions

Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen book series (GL, volume 162)

Abstract

The group of one-to-one conformai mappings of the extended plane onto itself is given analytically by the set of linear fractional transformations of the form
$$S\left( z \right) = \frac{{az + b}}{{cz + d}}\quad \left( {ad - bc \ne 0} \right) $$
(1.1)
.

Keywords

Unit Circle Boundary Point Unit Disk Conformal Mapping Fundamental Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    R. Nevanlinna [12].Google Scholar
  2. 2.
    One says that a Jordan are γ belonging to the boundary Γ is “free” provided none of its interior points is a point of accumulation of boundary points lying exterior to γ.Google Scholar
  3. 1.
    A. Speiser [2], R. Nevanlinna [10], G. Elfving [1].Google Scholar
  4. 1.
    Cf. F. Nevanlinna [2].Google Scholar
  5. 1.
    It the boundary point a ν is the point at infinity, za ν in the above expression has to be replaced by 1/z.Google Scholar
  6. 1.
    Even for this particular choice of the curves q ν, both “halves” of a fundamental region are in general not reflections of one another. One can show that this is the case only when all the points a ν lie on a circle.Google Scholar
  7. 2.
    Cf. e.g. C. Carathéodory [4].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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