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Existence of Perfect Picard Sets

Published online by Cambridge University Press:  22 January 2016

Kikuji Matsumoto
Kikuji Matsumoto*
Affiliation:
Mathematical Institute, Nagoya University
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Let E be a totally disconnected compact set in the z-plane and let Ω be its complement with respect to the extended 2-plane. Then Ω is a domain and we can consider a single-valued meromorphic function f(z) in Ω which has a transcendental singularity at each point ζ ∊ E. Suppose that E is a null-set of the class W in the sense of Kametani [4] (= the class NB in the sense of Ahlfors and Beurling [1]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 213 - 222
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Ahlfors, L. V. and Beurling, A.: Conformal invariants and function-theoretic null-sets, Acta Math., 83 (1950), pp. 101–129.CrossRefGoogle Scholar
[2] Bohr, H. and Landau, E.: Über das Verhalten von ζ(s) und ζk(s) in der Nähe der Geraden a = l, Göttinger Nachr., (1910).Google Scholar
[3] Carleson, L.: A remark on Picard’s theorem, Bull. Amer. Math. Soc, 67 (1961), pp. 142–144.CrossRefGoogle Scholar
[4] Kametani, S.: On Hausdorff’s measures and generalized capacities with some of their applications to the theory of functions, Jap. Journ. Math., 19 (1944–48), pp. 217–257.Google Scholar
[5] Lehto, O.: A generalization of Picard’s theorem, Ark. Mat., 3 nr. 45 (1958), pp. 495–500.CrossRefGoogle Scholar
[6] Matsumoto, K.: Exceptional values of meromorphic functions in a neighborhood of the set of singularities, Journ. Sci. Hiroshima Univ., A 24 (1960), pp. 143–153.Google Scholar
[7] Matsumoto, K.: On exceptional values of meromorphic functions with the set of singularities of capacity zero, Nagoya Math. Journ., 18 (1961), pp. 171–191.CrossRefGoogle Scholar
[8] Matsumoto, K.: Some notes on exceptional values of meromorphic functions, Nagoya Math. Journ., 22 (1963), pp. 189–201.CrossRefGoogle Scholar