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Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie. 3., erw. Aufl. (German) Zbl 0601.30001
Berlin etc.: Springer-Verlag. XI, 201 S. DM 96.00 (1986).
The first two editions appeared in 1916 and 1929 (see JFM 46.0469.02 and JFM 55.0171.03). It is a pleasure to see this book once more in print. Even though the results are hardly ”neuere” any longer, they still deserve to be read both for the charm of Landau’s exposition and their intrinsic interest. Many of them, simple as their statements often are, have become the starting points of extensive theories, which, for all their depth are perhaps less attractive than the sources from which they sprang. The theorems mainly deal with various phenomena presented by power series, especially with Tauberian theorems, singular points on the boundary, Picard’s theorem, and univalent functions, all still lively subjects today; the chapter on the equivalence of Cesàro and Hölder summability has proved to be less lasting. Professor Gaier has brought the book up to date by providing brief accounts of everything that has subsequently been done with each topic (the bibliography has 368 entries). In addition, he has presented full accounts of several modern germs: function theoretic proofs of Tauberian theorems, the proof of Fabry’s gap theorem by Turán’s lemma, Wermer’s maximality theorem, and the isomorphism theorem of Bers.
Reviewer: R.P.Boas

MSC:
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
40-02 Research exposition (monographs, survey articles) pertaining to sequences, series, summability