# Removable set

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff) of points of the complex plane for a certain class of functions analytic in a domain A compact set such that any function of class in can be continued as a function of class to the whole domain . 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: . It is assumed that the complement is a domain and that the class is defined for any domain.

According to another definition, a set is removable for a class , , if the fact that is a function of class in the complement implies that . Here the membership relations and are generally speaking not equivalent.

A first result on removable sets was the classical Cauchy–Riemann theorem on removable singularities: If a function is analytic and bounded in a punctured neighbourhood of a point , then it can be continued analytically to . A wider statement of the question (Painlevé's problem) is due to P. Painlevé: To find necessary and sufficient conditions on a set in order that , where is the class of all bounded analytic functions (cf. ). Painlevé himself found a sufficient condition: must have linear Hausdorff measure zero. Necessary and sufficient conditions for Painlevé's problem were obtained by L.V. Ahlfors (cf. ): if and only if has zero analytic capacity. There exists an example of a set 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 of several complex variables , , the statement of the problem on removable sets is changed by virtue of the classical Osgood–Brown theorem: If is a regular analytic function in a domain , except possibly on a compact set for which the complement is connected, then can be continued analytically to the whole domain . For other theorems on removable sets for , 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 be a domain in Euclidean space , , let be compact, , let be the class of bounded harmonic functions, and let be the class of harmonic functions with finite Dirichlet integral. Then the membership relations and are equivalent and are valid if and only if the capacity of is zero (cf. , ).

How to Cite This Entry:
Removable set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Removable_set&oldid=14994
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article