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Proceedings of the American Mathematical Society

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Bolzano's theorem in several complex variables


Author: Mau Hsiang Shih
Journal: Proc. Amer. Math. Soc. 79 (1980), 32-34
MSC: Primary 32H99
DOI: https://doi.org/10.1090/S0002-9939-1980-0560578-1
MathSciNet review: 560578
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Abstract: Let $ \Omega $ be a bounded domain in $ {C^n}$ containing the origin. Let $ f:\bar \Omega \to {C^n}$ be analytic in $ \Omega $ and continuous in $ \bar \Omega $, and $ \operatorname{Re} \bar z \cdot f(z) > 0$ for $ z \in \partial \Omega $. It is shown that f has exactly one zero in $ \Omega $.


References [Enhancements On Off] (What's this?)

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  • [2] Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR 0027863
  • [3] Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
  • [4] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR 0433481
  • [5] M. H. Shin, An analog of Bolzano's theorem for functions of a complex variable, Amer. Math. Monthly (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0560578-1
Keywords: Analytic functions, Bolzano's theorem, homotopy invariance, subvariety
Article copyright: © Copyright 1980 American Mathematical Society