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
On Cauchy’s Integral Theorem in the Plane, Proc. Nat. Acad. Sci. USA 25 (1939) 621–625.
On Green’s Formula, Proc. Nat. Acad. Sci. USA 26 (1940) 660–664.
The Behavior of a Complex Function at Infinity, Proc. Nat. Acad. Sci. USA 41 (1955) 512–513.
A Characterization of Weierstrass Analytic Functions, Proc. Nat. Acad. Sci. USA 54 (1965) 1025–1026.
Analytische Funktionen, Wiss. Abh. Forschungsgem. Nordrhein-Westfalen 33 (1965) 609–612. (= Festschrift Gedächtnisfeier K. Weierstrass, Westdeutscher Verlag, Köln, 1966).
Une charactérisation des fonctions analytiques, CRP 261 (1965) 4968–4969.
Weierstrass Analytic Functions, Math. Ann. 167 (1966) 177–194.
II. Further References
Ahlfors, L.: Complex Analysis, Second Edition. McGraw-Hill Book Company. New York 1966.
Bieberbach, L.: Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt. Grundlehren der mathematischen Wissenschaften Band LXVI, Springer Verlag, Berlin, 1953.
Bolza, O.: Lectures on the Calculus of Variations, Chicago, 1904.
Hille, E.: Ordinary Differential Equations in the Complex Domain, John Wiley, New York, 1976.
Kaup, L., Kaup, B.: Holomorphic Functions in Several Variables, Walter de Gruyter, Berlin, 1983.
Remmert, R.: Funktionentheorie, Dritte Auflage. Springer Verlag, Berlin, 1992.
Siegel, C. L.: Topics in Complex Function Theory, Vol. 1. Elliptic Functions and Uniformization Theory. Wiley-Interscience, New York, 1969.
Weyl, H.: Die Idee der Riemannschen Fläche, 5. Auflage, B.G. Teubner, Stuttgart, 1974.
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Reich, L. (2003). Commentary on Karl Menger’s Contributions to Analysis. In: Schweizer, B., et al. Selecta Mathematica. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6045-9_3
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