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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
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
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  • [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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Keywords: Analytic functions, Bolzano's theorem, homotopy invariance, subvariety
Article copyright: © Copyright 1980 American Mathematical Society

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