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Equilibrium points of logarithmic potentials on convex domains


Author: J. K. Langley
Journal: Proc. Amer. Math. Soc. 135 (2007), 2821-2826
MSC (2000): Primary 30D35, 31A05, 31B05
DOI: https://doi.org/10.1090/S0002-9939-07-08791-6
Published electronically: February 7, 2007
MathSciNet review: 2317957
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ D$ be a convex domain in $ \mathbb{C}$. Let $ a_k > 0$ be summable constants and let $ z_k \in D$. If the $ z_k$ converge sufficiently rapidly to $ \zeta \in \partial D$ from within an appropriate Stolz angle, then the function $ \sum_{k=1}^\infty a_k /( z - z_k ) $ has infinitely many zeros in $ D$. An example shows that the hypotheses on the $ z_k$ are not redundant and that two recently advanced conjectures are false.


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Additional Information

J. K. Langley
Affiliation: School of Mathematical Sciences, University of Nottingham, NG7 2RD, United Kingdom
Email: jkl@maths.nott.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-07-08791-6
Keywords: Critical points, potentials, zeros of meromorphic functions.
Received by editor(s): February 21, 2006
Received by editor(s) in revised form: May 19, 2006
Published electronically: February 7, 2007
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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