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Cluster Sets of Pseudomeromorphic Functions

Published online by Cambridge University Press:  22 January 2016

D. A. Storvick
D. A. Storvick*
Affiliation:
Department of Mathematics, University of Minnesota
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Let w = f(z) = u(x, y) + iv(x, y) be an interior transformation in the sense of Stoïlow in an arbitrary domain D, i.e. w = f(z) is continuous and single-valued in D, and unless constant takes open sets in D into open sets in the w-plane, and does not take any continuum (other than a single point) into a single point of the w-plane. Stoïlow proved that such an interior transformation can be represented as

where T(z) = ξ is a topological mapping of D onto a domain D′ and w = φ(ξ) is a meromorphic function in D′.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 18 , April 1961 , pp. 43 - 51
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1961

References

[1] Bagemihl, F., Curvilinear cluster sets of arbitrary functions, Proc. Nat. Acad. Sci. (U.S.A.), 41(1955), pp. 379–382.Google Scholar
[2] Carathéodory, C., Zuni Schwarzschen Spiegelungsprinzip, Comm. Math. Helv., 19 (1946–47), 263–278.Google Scholar
[3] Frostman, O., Potential d’équilibre et capacité des ensembles, Thèse Lund, (1935).Google Scholar
[4] Kametani, S.: The exceptional values of functions with the set of capacity zero of essential singularities, Proc. Acad. Tokyo, 17(1941), pp. 429–433.Google Scholar
[5] Lohwater, A. J., On the theorems of Gross and Iversen, ∧ir Force Report AFOSR TN 59–550, AD 216 660(1959), pp. 1–16.Google Scholar
[6] Mori, A., On quasi-conformality and pseudo-analyticity, Trans. Amer. Math. Soc, 84 (1957), pp. 56–77.CrossRefGoogle Scholar
[7] Nevanlinna, R., Eindeutige analytische Funktionen, 2nd ed. Berlin (1953).Google Scholar
[8] Noshiro, K., On the theory of cluster sets of analytic functions, Sugaku, 5(1953), pp. 65–72, AMS Translations, Series, 2, 8 (1958), pp. 1–12.Google Scholar
[9] Pfluger, A., Extremallängen und Kapazität, Comm. Math. Helv., 29 (1955), pp. 120–131.Google Scholar
[10] Stoïlow, S., Leçons sur les principes topologiques de la théorie des fonctions analytiques, Paris (1938).Google Scholar
[11] Storvick, D. A., On pseudo-analytic functions, Nagoya Math. Journal, Nagoya Math. Journal, 12(1957), pp. 131–138.CrossRefGoogle Scholar
[12] Tsuji, M., On the cluster set of a meromorphic function, Proc. Acad. Tokyo, 19 (1943).Google Scholar
[13] Tsuji, M., On the Riemann surface of an inverse function of a meromorphic function in the neighborhood of a closed set of capacity zero, Proc. Acad. Tokyo, 19(1943).Google Scholar
[14] Tsuji, M., Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, (1959).Google Scholar
[15] Whyburn, G. T., Analytic topology, Amer. Math. Soc. Colloq. Pub., 28(1942).Google Scholar
[16] Woolf, W. B.. Radial cluster sets and the distribution of values of meromorphic functions, Notices Amer, Math. Soc., Abstract, 562–19, 6(1959), p. 762.Google Scholar