Analytic Functions
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Analytic Functions Translated from the Second Gennan Edition by
Phillip Emig
Springer-Verlag Berlin Heidelberg GmbH 1970
Professor Dr. Rolf Nevanlinna Academy of Finland
Phillip Emig, Ph. D. Granada Hills, California (U.S.A.)
Gescbaftsfubrende Herausgeber:
Professor Dr. B. Eckmann Eidgenossische Technische Hocbschule Zurich
Professor Dr. B. L. van der Waerden Mathematischcs Institut der Universitat ZUrich
Revised translation of Eindeutige analytische Funktionen, 2nd ed. 1953 (Grundlehren der mathematischen Wissenschaften, Vol. 46)
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ISBN 978-3-642-85592-4 ISBN 978-3-642-85590-0 (eBook) DOI 10.1007/978-3-642-85590-0
© by Springer-Verlag Berlin Heidelberg 1970 Originally published by Springer-Verlag Berlin Heidelberg New York in 1970. Library of Congress Catalog Card Number 74·89810 Title No. 5145
Preface The present monograph on analytic functions coincides to a lar[ extent with the presentation of the modern theory of single-value analytic functions given in my earlier works "Le theoreme de Picarc Borel et la theorie des fonctions meromorphes" (Paris: Gauthier-Villar 1929) and "Eindeutige analytische Funktionen" (Die Grundlehren dt mathematischen Wissenschaften in Einzeldarstellungen, VoL 46, 1: edition Berlin: Springer 1936, 2nd edition Berlin-Gottingen-Heidelberg Springer 1953). In these presentations I have strived to make the individual result and their proofs readily understandable and to treat them in the ligh of certain guiding principles in a unified way. A decisive step in thi direction within the theory of entire and meromorphic functions consiste• in replacing the classical representation of these functions through ca nonical products with more general tools from the potential theor (Green's formula and especially the Poisson-Jensen formula). On thi foundation it was possible to introduce the quantities (the characteristic the proximity and the counting functions) which are definitive for th. description of the asymptotic properties of an analytic function in th( vicinity of essential singularities. At the same time they lead to far reaching extensions and sharpenings of Picard's theorem, in the directior of the general value distribution theory, which is concerned with the distribution and density of those points at which an analytic functior assumes a preassigned value, whereby all complex values are to b1 considered. This change in method has also led to new insights in anothe1 direction: it has helped to bring the algebraic-analytic points of viev. basic to the Cauchy-Weierstrass function theory into closer contact w
