The First Main Theorem in the Theory of Meromorphic Functions

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


In the present and in the following chapters we shall concern ourselves with the theory of those analytic functions that are of rational character, or as one expresses this more briefly, meromorphic at every point of a given schlicht region G. Such a function w = w(z) is hence regular in G except for poles; if the latter are infinite in number, they accumulate toward the boundary Γ of G. We shall further restrict ourselves to the simplest case, where G is simply connected; by the mono-dromy theorem, w(z) is then single-valued in G.


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  1. 1.
    Cf. T. Carleman [1], F. and R. Nevanlinna [1].Google Scholar
  2. 2.
    J. L.Jensen [1].Google Scholar
  3. 1.
    R. Nevanlinna [4].Google Scholar
  4. 1.
    It should be noted that except for the one value a = w(0), this counting function is identical with the function N(r, a) introduced on p. 53.Google Scholar
  5. 1.
    Here we use Landau’s notation O (φ(r)) for any quantity that is bounded when divided by φ(r).Google Scholar
  6. 1.
    T. Shimizu [1], L. Ahlfors [2]. A. Bloch [2] had earlier hinted at the possibility of such an interpretation.Google Scholar
  7. 1.
    Cf. L. Ahlfors [8].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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