Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-22T19:49:06.324Z Has data issue: false hasContentIssue false
  • English
  • Français

A Construction of Meromorphic Functions with Prescribed Asymptotic Behavior

Published online by Cambridge University Press:  22 January 2016

J.L. Stebbins
J.L. Stebbins*
Affiliation:
University of Wisconsin-Milwaukee Milwaukee, WisconsinU.S.A.
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.

Although there are several constructions of meromorphic functions with prescribed asymptotic sets [e.g., 5,6], it is usually difficult to determine or prescribe the nature of the asymptotic paths used in these constructions. On the other hand, there are several other constructions of meromorphic functions with prescribed asymptotic paths [e.g., 1, 10, 12], but the extent of the asymptotic values for these functions cannot always be restricted to the values approached along the given paths. Gross [3] has accomplished both results by prescribing paths for every value in the extended complex plane.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 41 , February 1971 , pp. 75 - 87
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Bagemihl, F., and Seidel, W.: Some boundary properties of analytic functions. -Math. Z. 61 (1954), 186–199.Google Scholar
[2] Barth, K.F.: Asymptotic values of meromorphic functions. -Michigan Math. J. 13 (1966), 321–340.Google Scholar
[3] Gross, W.: Eine ganze Funktion fur die jede komplexe zahl Konvergenzwert ist. -Math. Ann. 79 (1918), 201–208.Google Scholar
[4] Hausdorff, F.: Set Theory. -2nd english ed., Chelsea Publishing Co., 1962.Google Scholar
[5] Heins, M.: The set of asymptotic values of an entire function. -Scandinavian Mathematical Congress 12 (1953), 56–60.Google Scholar
[6] Kierst, S.: Sur l’ensemble des valeurs asymptotiques d1 une fonction méromorphe dans le cercle-unité. -Fund. Math. 27 (1936), 226–233.Google Scholar
[7] Kuratowski, K.: Topology I.-New edition, revised and augmented, Academic Press, 1966.Google Scholar
[8] MacLane, G.: Meromorphic functions with small characteristic and no asymptotic values. -Michigan Math. J. 8 (1961), 177–185.Google Scholar
[9] Mazurkiewicz, S.: Sur les points singuliers d’une fonction analytique. -Fund. Math. 17 (1931), 26–29.Google Scholar
[10] Roth, A.: Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen. -Comment. Math. Helv. 11 (1938), 77–125.Google Scholar
[11] Sierpinski, W.: General Topology. University of Toronto Press, 1962.Google Scholar
[12] Stebbins, J.L.: Spiral asymptotoc values of functions meromorphic in the unit disk. -Nagoya Math. J. 30 (1967), 247–262.Google Scholar
[13] Zalcman, L.: Analytic Capacity and Rational Approximation. -Springer-Verlog, Berlin, 1968.CrossRefGoogle Scholar