Selecta Mathematica: Volume 2 by Karl Menger (auth.), Prof. Dr. Bert Schweizer, Prof. Dr. Abe

By Karl Menger (auth.), Prof. Dr. Bert Schweizer, Prof. Dr. Abe Sklar, O. Univ.-Prof. Dr. Karl Sigmund, O. Univ.-Prof. Dr. DDr. h.c. Peter Gruber, em. O. Univ.-Prof. Dr. DDr. h.c. Edmund Hlawka, O. Univ.-Prof. Dr. Ludwig Reich, em. O. Univ.-Prof. Dr. Leopold

Karl Menger, one of many founders of size thought, belongs to the main unique mathematicians and thinkers of the 20th century. He was once a member of the Vienna Circle and the founding father of its mathematical similar, the Viennese Mathematical Colloquium. either in the course of his early years in Vienna and, after his emigration, within the usa, Karl Menger made major contributions to a wide selection of mathematical fields, and drastically inspired a few of his colleagues. The Selecta Mathematica include Menger's significant mathematical papers, in response to his personal choice from his vast writings. They care for themes as different as topology, geometry, research and algebra, in addition to writings on economics, sociology, good judgment, philopsophy and mathematical effects. the 2 volumes are a monument to the range and originality of Menger's ideas.

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Since for an inscribed polygon P the complementary domains approach those of C as P gets indefinitely dense in C, and tim p-p(p) = p-c(p) for each point p not on C, we get the theorem from that about closed polygons by the method used at the end of section 3. 5. -The method of proving the formula for a rectangular domain as the limit of an identity, and for more -44- 664 MATHEMATICS: G. C. EVANS PROC. N. A. S. complicated domains by virtue of the elementary continuity properties of the integrals can be applied to many formulae of vector analysis.

Menger emphasizes that for many decades this was the only exact alternative to introducing functions as "Iaws or mIes associating or pairing numbers with numbers" and multifunctions as mies 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 indudes, in contrast to Weierstrass' complete analytic functions, also poles and algebraic ramification points (see also C.

A5. Analytische Funktionen, Wiss. Abh. Forschungsgern. Nordrhein-Westfalen 33 (1965) 609-612. (= Festschrift Gedächtnisfeier K. Weierstrass, Westdeutscher Verlag, Köln, 1966). A6. Une characterisation des fonctions analytiques, CRP 261 (1965) 4968-4969. 7. Weierstrass Analytic Functions, Math. Ann. 167 (1966) 177-194. -33- Ludwig Reich II. Further References Ahlfors. : Complex Analysis, Second Edition. McGraw-Hill Book Company. New York 1966. : Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt.

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