# Polar set

The polar set of analytic function $f(z)$ of the complex variable $z=(z_1,\dots,z_n)$, $n\geq1$, is the set $P$ of points in some domain $D$ of the complex space $\mathbf C^n$ for which: a) $f(z)$ is holomorphic everywhere in $D\setminus P$; b) $f(z)$ cannot be analytically continued to any point of $P$; and c) for every point $a\in P$ there are a neighbourhood $U_a$ and a function $q_a(z)\not\equiv0$, holomorphic in $U_a$, for which the function $p_a(z)=q_a(z)f(z)$, which is holomorphic in $D\cap\{U_a\setminus P\}$, can be holomorphically continued to $U_a$. At every point $a\in P$ one has $q_a(a)=0$. The polar set $P$ consists of the poles (cf. Pole (of a function)) $a\in P$ of $f(z)$, for which $p_a(a)\neq0$, and the points $a\in P$ of indeterminacy of $f(z)$, for which $p_a(a)=0$ (it is assumed that $p_a(z)$ and $q_a(z)$ have no common factors that are holomorphic and vanish at $a$). Every polar set is a complex analytic variety (by which we mean the set of common zeros of a finite set of holomorphic functions) of complex dimension $n-1$.

A polar set in potential theory is a set $E$ of points of the Euclidean space $\mathbf R^n$, $n\geq2$, for which there exists a potential $U_\mu(x)$, $x\in\mathbf R^n$, for some Borel measure $\mu$, that takes the value $+\infty$ at the points of $E$ and only at those points.

In the case of the logarithmic potential for $n=2$ and the Newton potential for $n\geq3$, for a bounded set $E$ to be a polar set it is necessary and sufficient that $E$ is a set of type $G_\delta$ and has zero outer capacity. Here, in the definition of a polar set, one can replace "potential" by "superharmonic function". The main properties of polar sets in this case are: a) the set $\{a\}$ which consists of a single point $a\in\mathbf R^n$ is a polar set; b) a countable union of polar sets is a polar set; c) any polar set has Lebesgue measure zero in $\mathbf R^n$; and d) under a conformal mapping a polar set goes to a polar set.

For a local criterion for being polar see Thinness of a set.

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How to Cite This Entry:
Polar set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_set&oldid=42176
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article