Equilibrium points of logarithmic potentials on convex domains

Langley, J.

doi:10.1090/S0002-9939-07-08791-6

Equilibrium points of logarithmic potentials on convex domains
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by J. K. Langley PDF

Proc. Amer. Math. Soc. 135 (2007), 2821-2826 Request permission

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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