# Removable set

$E$ 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. [1]). 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. [2]): $E \in N ( A B , G )$ 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 [3], [4], [6], [9].

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. [7], [10].

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. [5], [8]).

#### References

 [1] L. Zoretti, "Leçons sur la prolongement analytique" , Gauthier-Villars (1911) [2] L.V. Ahlfors, "Bounded analytic functions" Duke Math. J. , 14 : 1 (1947) pp. 1–11 MR0021108 Zbl 0030.03001 [3] K. Nohiro, "Cluster sets" , Springer (1960) [4] S.Ya. Khavinson, "Analytic functions of bounded type" Itogi Nauk. Mat. Anal. 1963 (1965) pp. 5–80 (In Russian) [5] L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967) MR0225986 Zbl 0189.10903 [6] M.S. Mel'nikov, S.O. Sinanyan, "Aspects of approximation theory for functions of one complex variable" J. Soviet Math. , 5 : 5 (1976) pp. 688–752 Itogi Nauk. i Tekhn. Sovremen. Probl. Mat. , 4 (1975) pp. 143–250 Zbl 0375.30014 [7] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1985) (In Russian) Zbl 0578.32001 Zbl 0574.30001 [8] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976) MR0460672 MR0419791 MR0412442 MR0442324 Zbl 0419.31001 Zbl 0339.31003 Zbl 0328.33011 [9] E.P. Dolzhenko, "Elimination of singularities of analytic functions" Uspekhi Mat. Nauk , 18 : 4 (1963) pp. 135–142 (In Russian) [10] L.J. Riihentaus, "Removable singularities of analytic functions of several complex variables" Math. Z. , 158 (1978) pp. 45–54 MR0484286 Zbl 0351.32014

For quite general continuation results see [a2]; Riemann's theorem has an analogue in $\mathbf C ^ {n}$: Bounded analytic functions extend analytically across subvarieties of codimension $\geq 1$, while all analytic functions can be analytically continued across subvarieties of codimension $\geq 2$. See [a1].