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Commentary on Karl Menger’s Contributions to Analysis

  • Ludwig Reich
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Abstract

Let \(P(z) = {{w}_{0}} + \sum _{1}^{n}{{a}_{n}}{{(z - {{z}_{0}})}^{n}}\) be a power series with complex coefficients and with a radius of convergence different from 0. Then K. Weierstrass introduced the notion of “analytisches Gebilde” (complete analytic function) defined by P as the set of all power series \({{w}_{1}} + \sum _{1}^{n}a_{n}^{{(1)}}{{(z - {{z}_{1}})}^{n}}\) obtained from P by direct and indirect analytic (i.e., holomorphic) continuation. In his article A.7, which is supplemented by his papers A.4, A.5 and A.6, K. Menger emphasizes that for many decades this was the only exact alternative to introducing functions as “laws or rules associating or pairing numbers with numbers” and multifunctions as rules of pairing numbers with sets of numbers. It is well-known today that Weierstrass’ notion of a complete analytic function leads in a natural way to the concept of “analytisches Gebilde” as given by H. Weyl in his famous book [W], §2, §3. This again is, after introducing an appropriate natural topology, a nontrivial example of a Riemann surface, and it includes, in contrast to Weierstrass’ complete analytic functions, also poles and algebraic ramification points (see also C. L. Siegel’s lectures [S], Chapter 1, 3, Chapter 1, 4). It is also well-known to mathematicians today that the definition of Riemann surfaces as a class of two-dimensional manifolds satisfying a certain regularity condition involves the use of a class of changes of the local parameters (coordinates), namely exactly those which are given by locally biholomorphic functions.

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References

I. Selected Papers of K. Menger on Analysis

  1. A.1
    On Cauchy’s Integral Theorem in the Plane, Proc. Nat. Acad. Sci. USA 25 (1939) 621–625.CrossRefGoogle Scholar
  2. A.2
    On Green’s Formula, Proc. Nat. Acad. Sci. USA 26 (1940) 660–664.CrossRefGoogle Scholar
  3. A.3
    The Behavior of a Complex Function at Infinity, Proc. Nat. Acad. Sci. USA 41 (1955) 512–513.CrossRefGoogle Scholar
  4. A.4
    A Characterization of Weierstrass Analytic Functions, Proc. Nat. Acad. Sci. USA 54 (1965) 1025–1026.CrossRefGoogle Scholar
  5. A.5
    Analytische Funktionen, Wiss. Abh. Forschungsgem. Nordrhein-Westfalen 33 (1965) 609–612. (= Festschrift Gedächtnisfeier K. Weierstrass, Westdeutscher Verlag, Köln, 1966).Google Scholar
  6. A.6
    Une charactérisation des fonctions analytiques, CRP 261 (1965) 4968–4969.Google Scholar
  7. A.7
    Weierstrass Analytic Functions, Math. Ann. 167 (1966) 177–194.MathSciNetCrossRefGoogle Scholar

II. Further References

  1. [A]
    Ahlfors, L.: Complex Analysis, Second Edition. McGraw-Hill Book Company. New York 1966.zbMATHGoogle Scholar
  2. [Bie]
    Bieberbach, L.: Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt. Grundlehren der mathematischen Wissenschaften Band LXVI, Springer Verlag, Berlin, 1953.CrossRefGoogle Scholar
  3. [Bo]
    Bolza, O.: Lectures on the Calculus of Variations, Chicago, 1904.zbMATHGoogle Scholar
  4. [H]
    Hille, E.: Ordinary Differential Equations in the Complex Domain, John Wiley, New York, 1976.zbMATHGoogle Scholar
  5. [K]
    Kaup, L., Kaup, B.: Holomorphic Functions in Several Variables, Walter de Gruyter, Berlin, 1983.CrossRefGoogle Scholar
  6. [R]
    Remmert, R.: Funktionentheorie, Dritte Auflage. Springer Verlag, Berlin, 1992.Google Scholar
  7. [S]
    Siegel, C. L.: Topics in Complex Function Theory, Vol. 1. Elliptic Functions and Uniformization Theory. Wiley-Interscience, New York, 1969.zbMATHGoogle Scholar
  8. [W]
    Weyl, H.: Die Idee der Riemannschen Fläche, 5. Auflage, B.G. Teubner, Stuttgart, 1974.zbMATHGoogle Scholar

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© Springer-Verlag Wien 2003

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  • Ludwig Reich

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