Conformal Mapping of Simply and Multiply Connected Regions

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


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
$$S\left( z \right) = \frac{{az + b}}{{cz + d}}\quad \left( {ad - bc \ne 0} \right) $$


Unit Circle Boundary Point Unit Disk Conformal Mapping Fundamental Theorem 
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  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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