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Cluster Sets

  • Book
  • © 1960

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Authors:
  1. Kiyoshi Noshiro
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Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (MATHE2, volume 28)

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Table of contents (4 chapters)

  1. Front Matter

    Pages II-VII
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  2. Definitions and preliminary discussions

    • Kiyoshi Noshiro
    Pages 1-5

  3. Single-valued analytic functions in general domains

    • Kiyoshi Noshiro
    Pages 5-31

  4. Functions meromorphic in the unit circle

    • Kiyoshi Noshiro
    Pages 32-90

  5. Conformal mapping of Riemann surfaces

    • Kiyoshi Noshiro
    Pages 90-109

  6. Back Matter

    Pages 109-135
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Keywords

About this book

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.

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