03518nmm a2200289 u 4500001001200000003002700012005001700039007002400056008004100080020001800121100002100139245006200160250001700222260006300239300004200302505107600344653002601420653001301446653002701459710003401486041001901520989003801539490010601577856007201683082000801755520146501763EB000677620EBX0100000000000000053070200000000000000.0cr|||||||||||||||||||||140122 ||| eng a97836428592811 aNoshiro, Kiyoshi00aCluster SetshElektronische Ressourcecby Kiyoshi Noshiro a1st ed. 1960 aBerlin, HeidelbergbSpringer Berlin Heidelbergc1960, 1960 aVIII, 136 p. 1 illusbonline resource0 aI. 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 aMathematical analysis aAnalysis aAnalysis (Mathematics)2 aSpringerLink (Online service)07aeng2ISO 639-2 bSBAaSpringer Book Archives -20040 aErgebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics uhttps://doi.org/10.1007/978-3-642-85928-1?nosfx=yxVerlag3Volltext0 a515 aFor 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