The Behavior of a Complex Function at Infinity

  • Karl Menger


Traditionally, the behavior at ∞ of a complex function f is defined as the behavior at 0 of the function obtained by substituting the —1st power into f. This definition adequately describes the class of values f(z) for large z. For instance, the range near ∞of the identity function j (whose value for any z is j(z)=z) coincides with the range near 0 of the function j-1. But that definition does not in any way describe the structural behavior of f near ∞, reflected in properties of the class of pairs (z, f (z)) for large z. In fact, the association of the value f(z) with z may, by the substitution of j-1, completely change its character. For instance, the derivative of j near ∞ is the constant function 1, whereas that j-1 of near 0 goes even faster to ∞ than does j-1

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Karl Menger
    • 1
  1. 1.Illinois Institute of TechnologyUSA

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