Commentary on Karl Menger’s Contributions to Analysis

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.