Bolzano’s theorem in several complex variables
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- by Mau Hsiang Shih PDF
- Proc. Amer. Math. Soc. 79 (1980), 32-34 Request permission
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
- Tom M. Apostol, Mathematical analysis, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974. MR 0344384
- Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR 0027863
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, 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. MR 0433481 M. H. Shin, An analog of Bolzano’s theorem for functions of a complex variable, Amer. Math. Monthly (to appear).
Additional Information
- © Copyright 1980 American Mathematical Society
- 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