# Removable set

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$E$ of points of the complex plane $\mathbf C$ for a certain class $K$ of functions analytic in a domain $G \subset \mathbf C$

A compact set $E \subset G$ such that any function $f ( z)$ of class $K$ in $G \setminus E$ can be continued as a function of class $K$ to the whole domain $G$. The situation may be described in other words by saying that "the set E is set, removable for a class of functionsremovable for the class K" or that "E is a null-set for the class K" , briefly: $E \in N ( K , G )$. It is assumed that the complement $G \setminus E$ is a domain and that the class $K$ is defined for any domain.

According to another definition, a set $E$ is removable for a class $K$, $E \in N ( K)$, if the fact that $f ( z)$ is a function of class $K$ in the complement $\mathbf C \setminus E$ implies that $f ( z) = \textrm{ const }$. Here the membership relations $E \in N ( K , G )$ and $E \in N ( K )$ are generally speaking not equivalent.

A first result on removable sets was the classical Cauchy–Riemann theorem on removable singularities: If a function $f ( z)$ is analytic and bounded in a punctured neighbourhood $V ( a) = \{ {z } : {0 < | z - a | < \delta } \}$ of a point $a \in \mathbf C$, then it can be continued analytically to $a$. A wider statement of the question (Painlevé's problem) is due to P. Painlevé: To find necessary and sufficient conditions on a set $E$ in order that $E \in N ( A B , G )$, where $K = A B$ is the class of all bounded analytic functions (cf. ). Painlevé himself found a sufficient condition: $E$ must have linear Hausdorff measure zero. Necessary and sufficient conditions for Painlevé's problem were obtained by L.V. Ahlfors (cf. ): $E \in N ( A B , G )$ if and only if $E$ has zero analytic capacity. There exists an example of a set $E$ of positive length but zero analytic capacity. On removable sets for different classes of analytic functions of one complex variable and related unsolved problems see , , , .

In the case of analytic functions $f ( z)$ of several complex variables $z = ( z _ {1} \dots z _ {n} )$, $n \geq 2$, the statement of the problem on removable sets is changed by virtue of the classical Osgood–Brown theorem: If $f ( z)$ is a regular analytic function in a domain $G \subset \mathbf C ^ {n}$, except possibly on a compact set $E \subset G$ for which the complement $G \setminus E$ is connected, then $f ( z)$ can be continued analytically to the whole domain $G$. For other theorems on removable sets for $n \geq 2$, as well as connections with the concept of a domain of holomorphy, see e.g. , .

The problem of removable sets can also be posed for harmonic, subharmonic and other functions. E.g., let $G$ be a domain in Euclidean space $\mathbf R ^ {n}$, $n \geq 3$, let $E$ be compact, $E \subset G$, let $H B$ be the class of bounded harmonic functions, and let $H D$ be the class of harmonic functions with finite Dirichlet integral. Then the membership relations $E \in N ( H B , G )$ and $E \in N ( H D , G )$ are equivalent and are valid if and only if the capacity of $E$ is zero (cf. , ).

How to Cite This Entry:
Removable set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Removable_set&oldid=48508
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article