Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T07:18:41.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Exceptionally ramified meromorphic functions with a non-enumerable set ofessential singularities

Published online by Cambridge University Press:  22 January 2016

Toshiko Kurokawa
Toshiko Kurokawa*
Affiliation:
Department of Mathematics, Mie University, Kamihama, Tsu-shi, 514, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the complex function theory, Picard’s Great Theorem plays an essential and important role. It is well-known as generalizations of this theorem that in a neighborhood of an isolated essential singularity, a meromorphic function cannot be exceptionally ramified (see W. Gross [2]) and that even it cannot be normal (see O. Lehto and K. I. Virtanen [7]). We are therefore interested in the behaviour of meromorphic functions with non-isolated essential singularities as well as in generalizations of the Gross’ result. Several approaches in this direction have been made by G. af Hällström [3], S. Kametani [4], K. Noshiro [13], K. Matsumoto [8], [9], [10], [11], [12], S. Toppila [15], etc..

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 88 , December 1982 , pp. 133 - 154
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[ 1 ] Besicovitch, A. S., On sufficient conditions for a function to be analytic, and on behaviour of analytic functions in the neighbourhood of nonisolated singular points, Proc. Lond. Math. Soc., Ser. 2, 32 (1930), 1–9.Google Scholar
[ 2 ] Gross, W., Über die Singularitäten analytischer Funktionen, Monat. Math. Physik, 29 (1918), 3–47.Google Scholar
[ 3 ] Hällström, G. af, Über meromorphe Funktionen mit mehrfach zusammenhängenden Existenzgebieten, Acta Acad. Abo. Math. Phys. 12 No.8 (1939), 100 pp.Google Scholar
[ 4 ] Kametani, S., The exceptional values of functions with the set of capacity zero of essential singularities I, II, Proc. Imp. Acad. Tokyo, 17 (1941), 429–433.Google Scholar
[ 5 ] Kurokawa, T. and Matsumoto, K., On totally ramified values, Töhoku Math. J., 28 (1976), 245–256.Google Scholar
[ 6 ] Lehto, O. and Virtanen, K. I., Quasiconformal mapplngs in the plane, Springer Verlag, Berlin-Heidelberg-New York, 1973.Google Scholar
[ 7 ] Lehto, O. and Virtanen, K. I., Boundary behaviour and normal meromorphic functions, Acta Math., 97 (1957), 47–65.Google Scholar
[ 8 ] Matsumoto, K., Exceptional values of meromorphic functions in a neighborhood of the set of singularities, J. Sci. Hiroshima Univ. Ser. A, 24 (1960), 143–153.Google Scholar
[ 9 ] Matsumoto, K., Some notes on exceptional values of meromorphic functions, Nagoya Math. J., 22 (1963), 189–201.Google Scholar
[10] Matsumoto, K., Existence of perfect Picard sets, Nagoya Math. J., 27 (1966), 213–222.Google Scholar
[11] Matsumoto, K., Perfect Picard set of positive capacity, Nagoya Math. J., 29 (1967), 51–56.Google Scholar
[12] Matsumoto, K., Some remarks on Picard sets, Ann. Acad. Sci. Fennicae, Ser. A. I., 403 1967).Google Scholar
[13] Noshiro, K., Contributions to the theory of the singularities of analytic functions, Japan. J. Math. 19 No. 4 (1948), 299–327.Google Scholar
[14] Sario, L. and Noshiro, K., Value distribution theory, D. Van Nostrand Comp., Inc., Toronto-New York-London, 1966.Google Scholar
[15] Toppila, S., Picard sets for meromorphic functions, Ann. Acad. Sci. Fennicae, A. I. 417 (1967).Google Scholar