Geometric theory of functions of a complex variable

Abstract

Vorlesungen ilber Funktionentheorie. By A. Dinghas. Springer Verlag, Berlin, 1961. 15+403 pp. Currently the collection of textbooks and monographs that aim to present the classical theory of functions of a single complex variable and possibly aspects of its subsequent development is vast. This remark is not a complaint. On the contrary. The teacher who is concerned with the theory of functions of a complex variable has never been in a more favorable situation as far as the availability of a rich and varied collection of monographs and textbooks is concerned. This is a phenomenon of the last fifteen years or so. There were wellestablished texts in the period preceding World War II—such as the Lehrbücher of Osgood, Hurwitz-Courant, and Bieberbach, the Cours and Traités of Paris, the treatises of Dienes, Titchmarsh, and Saks-Zygmund (available then only in Polish but of recognized importance in spite of the linguistic barrier), the celebrated monographs of R. Nevanlinna, the Borel series, the "Aufgaben und Lehrsatze" of Pólya and Szegö, and Landau's "Neuere Ergebnisse." But it was inevitable that the burgeoning research activity in the theory of functions of a complex variable which has taken place in the last three decades should at last find itself reported in the monographic literature. In this connection we cite two treatises of major size: Golusin's Geometric theory of functions of a complex variable now available to a large group of readers through its translation into German, and Tsuji's Potential theory in modern function theory, both directed to students who have gone beyond the elements. I t is by keeping in mind the historical situation we have described that we can appraise correctly the special character of the Vorlesungen iiber Funktionentheorie of Professor A. Dinghas. Its goal is an exposé d'ensemble of the classical theory of functions of a single complex variable together with a number of its principal modern developments, especially those of the last three decades, beginning with first principles and advancing to the frontiers of modern investigations. Some idea of the magnitude of such an undertaking may be had when we realize that the author has not contented himself to present the classical groundwork of the subject in the old traditional manner, but has taken into consideration the reworking of the fundamentals by such mathematicians as Artin (Notre Dame Monograph) and Ahlfors (Complex Analysis). As is well known, the homology-theoretic exposition of the classical Cauchy theory given by Ahlfors has left its mark and has brought to the fore a point of view that no writer of a modern text on the theory of functions of a complex variable can pos-