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＜E-Book＞
Cluster Sets / by Kiyoshi Noshiro
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics ; 28)

Edition	1st ed. 1960.
Publisher	(Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer)
Year	1960
Size	VIII, 136 p. 1 illus : online resource
Authors	*Noshiro, Kiyoshi author SpringerLink (Online service)
Subjects	LCSH:Mathematical analysis FREE:Analysis
Notes	I. Definitions and preliminary discussions -- § 1. Definitions of cluster sets -- § 2. Some classical theorems -- II. Single-valued analytic functions in general domains -- § 1. Compact set of capacity zero and Evans-Selberg’s theorem -- § 2. Meromorphic functions with a compact set of essential singularities of capacity zero -- § 3. Extension of Iversen’s theorem on asymptotic values -- § 4. Extension of Iversen-Gross-Seidel-Beurling’s theorem -- § 5. Hervé’s theorems -- III. Functions meromorphic in the unit circle -- §1. Functions of class (U) in Seidel’s sense -- § 2. Boundary theorems of Collingwood and Cartwright -- § 3. Baire category and cluster sets -- § 4. Boundary behaviour of meromorphic functions -- § 5. Meromorphic functions of bounded type and normal meromorphic functions -- IV. Conformal mapping of Riemann surfaces -- § 1. Gross’ property of covering surfaces -- § 2. Iversen’s property of covering surfaces -- § 3. Boundary theorems on open Riemann surfaces -- Appendix: Cluster sets of pseudo-analytic functions For the first systematic investigations of the theory of cluster sets of analytic functions, we are indebted to IVERSEN [1-3J and GROSS [1-3J about forty years ago. Subsequent important contributions before 1940 were made by SEIDEL [1-2J, DOOE [1-4J, CARTWRIGHT [1-3J and BEURLING [1]. The investigations of SEIDEL and BEURLING gave great impetus and interest to Japanese mathematicians; beginning about 1940 some contributions were made to the theory by KUNUGUI [1-3J, IRIE [IJ, TOKI [IJ, TUMURA [1-2J, KAMETANI [1-4J, TsuJI [4J and NOSHIRO [1-4J. Recently, many noteworthy advances have been made by BAGEMIHL, SEIDEL, COLLINGWOOD, CARTWRIGHT, HERVE, LEHTO, LOHWATER, MEIER, OHTSUKA and many other mathematicians. The main purpose of this small book is to give a systematic account on the theory of cluster sets. Chapter I is devoted to some definitions and preliminary discussions. In Chapter II, we treat extensions of classical results on cluster sets to the case of single-valued analytic functions in a general plane domain whose boundary contains a compact set of essential singularities of capacity zero; it is well-known that HALLSTROM [2J and TsuJI [7J extended independently Nevanlinna's theory of meromorphic functions to the case of a compact set of essential singUlarities of logarithmic capacity zero. Here, Ahlfors' theory of covering surfaces plays a funda­ mental role. Chapter III "is concerned with functions meromorphic in the unit circle HTTP:URL=https://doi.org/10.1007/978-3-642-85928-1

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