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

Comments

The Osgood–Brown theorem is also called Hartogs' theorem, cf. Hartogs theorem.

See also Analytic continuation; Analytic set; Removable singular point.

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].

References

[a1]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Section G MR0180696 Zbl 0141.08601
[a2]	R. Harvey, J. Polking, "Removable singularities of solutions of linear partial differential equations" Acta Math. , 125 (1970) pp. 39–55 MR0279461 Zbl 0214.10001
[a3]	J.B. Garnett, "Analytic capacity and measure" , Lect. notes in math. , 297 , Springer (1972) MR0454006 Zbl 0253.30014
[a4]	E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) MR1111477 Zbl 0683.32002