Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T04:17:28.609Z Has data issue: false hasContentIssue false
  • English
  • Français

On Analytic Functions on Some Riemann Surfaces

Published online by Cambridge University Press:  22 January 2016

Tadashi Kuroda
Tadashi Kuroda*
Affiliation:
Mathematical Institute Nagoya University
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 theory of functions meromorphic in |z| < + ∞, Iversen [4] proved the following: If w = f(z) is meromorphic in |z| < + ∞ and has an essential singularity at z = ∞, then any inverse function-element of this function with the centre w0 can be continued analytically to any point w ≠ wo, except possibly this point w, in any disc having the centre at the point w and containing the point w0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 10 , June 1956 , pp. 27 - 50
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1956

References

[ 1 ] Ahlfors, L.-Beurling, A. : Conformal invariants and function-theoretic null sets, Acta Math., 83 (1950), 101–129.Google Scholar
[ 2 ] Bader, R.-Parreau, M. : Domaines non-compacts et classification des surfaces de Riemann, C. R. Acad. Sci. Paris, 232 (1951), 138–139.Google Scholar
[ 3 ] Heins, M.: On the Lindelöf principle, Ann. of Math., 61 (1955), 440–473.Google Scholar
[ 4 ] Iversen, F. : Recherches sur les fonctions inverses des fonctions meromorphes, Thèse de Helsingfors (1914).Google Scholar
[ 5 ] Kametani, S. : On Hausdorff’s measure and generalized capacities with some of their applications to the theory of functions, Jap. Journ. Math., 19 (1945-48), 217–257.CrossRefGoogle Scholar
[ 6 ] Kuramochi, Z. : On covering property of abstract Riemann surfaces, Osaka Math. Journ., 6 (1954), 93–103.Google Scholar
[ 7 ] Kuramochi, Z. : On the behaviour of analytic functions on abstract Riemann surfaces, Osaka Math. Journ., 7 (1955), 109–127.Google Scholar
[ 8 ] Kuroda, T. : A property of some open Riemann surfaces and its application, Nagoya Math. Journ., 6 (1953), 77–84.Google Scholar
[ 9 ] Kuroda, T. : Theorems of the Phragmén-Lindelöf type on an open Riemann surface, Osaka Math. Journ., 6 (1954), 231–241.Google Scholar
[10] Mori, A. : On Riemann surfaces, on which no bounded harmonic function exists, Journ. Math. Soc. Jap., 3 (1951), 285–289.Google Scholar
[11] Mori, A. : On the existence of harmonic functions on a Riemann surface, Journ. Fac. Sci. Univ. Tokyo, S. I., 6 (1951), 247–257.Google Scholar
[12] Mori, A. : A note on unramified abelian covering surfaces of a closed Riemann surface, Journ. Math. Soc. Jap., 6 (1954), 162–176.Google Scholar
[13] Myrberg, P. J. : Die Kapazität der singulären Menge der linearen Gruppen, Ann. Acad. Sci. Fenn., A. I. 10 (1941).Google Scholar
[14] Myrberg, P. J. : Über die analytische Fortsetzung von beschränkten Funktionen, Ann. Acad. Sci. Fenn., A. I. 58 (1949).Google Scholar
[15] Nevanlinna, R. : Über die Existenz von beschränkten Potentialfunktionen auf Flachen von unendlichen Geschlecht, Math. Zeit, 52 (1950), 559–604.CrossRefGoogle Scholar
[16] Noshiro, K.: Open Riemann surface with null boundary, Nagoya Math. Journ., 3 (1951), 73–79.Google Scholar
[17] Pfluger, A. : Über das Anwachsen eindeutiger analytischer Funktionen auf offenen Riemannschen Flächen, Ann. Acad. Sci. Fenn., A. I. 64 (1949).Google Scholar
[18] Pfluger, A. : Sur l’existence de fonctions non constantes, analytiques, uniformes et bornées sur une surface de Riemann ouverte, C. R. Acad. Sci. Paris, 230 (1950), 166–168.Google Scholar
[19] Sario, L. : Über Riemannsche Flächen mit hebbarem Rand, Ann. Acad. Sci. Fenn., A. I. 50 (1948).Google Scholar
[20] Sario, L. : Sur la classification des surfaces de Riemann, 11 Congr. Math. Scand., (1949), 229–238.Google Scholar
[21] Sario, L. : Modular criterion on Riemann surfaces, Duke Math. Journ., 20 (1953), 279–286.Google Scholar
[22] Stoïlow, S. : Sur les singularités des fonctions analytiques multiformes dont la surface de Riemann a sa frontière de mesure harmonique nulle, Mathematica, Timisoara, 19 (1943), 126–138.Google Scholar
[23] Tôki, Y. : On the classification of open Riemann surfaces, Osaka Math. Journ., 4 (1952), 191–202.Google Scholar
[24] Tôki, Y.: On the examples in the classification of open Riemann surfaces (I), Osaka Math. Journ., 5 (1953), 267–280.Google Scholar
[25] Tsuji, M. : Theory of meromorphic functions on an open Riemann surface with null boundary, Nagoya Math. Journ., 6 (1953), 137–150.Google Scholar
[26] Virtanen, K. I. : Über die Existenz von beschrankten harmonischen Funktionen auf offenen Riemannschen Flächen, Ann. Acad. Sci. Fenn., A. I. 75 (1950).Google Scholar