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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812301.png" /> of points of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812302.png" /> for a certain class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812303.png" /> of functions analytic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812304.png" />''
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A compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812305.png" /> such that any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812306.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812307.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812308.png" /> can be continued as a function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r0812309.png" /> to the whole domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123010.png" />. 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123011.png" />. It is assumed that the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123012.png" /> is a domain and that the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123013.png" /> is defined for any domain.
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According to another definition, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123014.png" /> is removable for a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123016.png" />, if the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123017.png" /> is a function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123018.png" /> in the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123019.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123020.png" />. Here the membership relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123022.png" /> are generally speaking not equivalent.
+
'' $  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 first result on removable sets was the classical Cauchy–Riemann theorem on removable singularities: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123023.png" /> is analytic and bounded in a punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123024.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123025.png" />, then it can be continued analytically to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123026.png" />. A wider statement of the question (Painlevé's problem) is due to P. Painlevé: To find necessary and sufficient conditions on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123027.png" /> in order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123029.png" /> is the class of all bounded analytic functions (cf. [[#References|[1]]]). Painlevé himself found a sufficient condition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123030.png" /> must have linear Hausdorff measure zero. Necessary and sufficient conditions for Painlevé's problem were obtained by L.V. Ahlfors (cf. [[#References|[2]]]): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123031.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123032.png" /> has zero [[Analytic capacity|analytic capacity]]. There exists an example of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123033.png" /> 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 [[#References|[3]]], [[#References|[4]]], [[#References|[6]]], [[#References|[9]]].
+
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.
  
In the case of analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123034.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123036.png" />, the statement of the problem on removable sets is changed by virtue of the classical Osgood–Brown theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123037.png" /> is a regular analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123038.png" />, except possibly on a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123039.png" /> for which the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123040.png" /> is connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123041.png" /> can be continued analytically to the whole domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123042.png" />. For other theorems on removable sets for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123043.png" />, as well as connections with the concept of a [[Domain of holomorphy|domain of holomorphy]], see e.g. [[#References|[7]]], [[#References|[10]]].
+
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.
  
The problem of removable sets can also be posed for harmonic, subharmonic and other functions. E.g., let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123044.png" /> be a domain in Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123046.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123047.png" /> be compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123048.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123049.png" /> be the class of bounded harmonic functions, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123050.png" /> be the class of harmonic functions with finite Dirichlet integral. Then the membership relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123052.png" /> are equivalent and are valid if and only if the [[Capacity|capacity]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123053.png" /> is zero (cf. [[#References|[5]]], [[#References|[8]]]).
+
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. [[#References|[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. [[#References|[2]]]):  $  E \in N ( A B , G ) $
 +
if and only if  $  E $
 +
has zero [[Analytic capacity|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 [[#References|[3]]], [[#References|[4]]], [[#References|[6]]], [[#References|[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|domain of holomorphy]], see e.g. [[#References|[7]]], [[#References|[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|capacity]] of $  E $
 +
is zero (cf. [[#References|[5]]], [[#References|[8]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Zoretti, "Leçons sur la prolongement analytique" , Gauthier-Villars (1911)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.V. Ahlfors, "Bounded analytic functions" ''Duke Math. J.'' , '''14''' : 1 (1947) pp. 1–11 {{MR|0021108}} {{ZBL|0030.03001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> K. Nohiro, "Cluster sets" , Springer (1960)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S.Ya. Khavinson, "Analytic functions of bounded type" ''Itogi Nauk. Mat. Anal. 1963'' (1965) pp. 5–80 (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967) {{MR|0225986}} {{ZBL|0189.10903}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0375.30014}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''2''' , Moscow (1985) (In Russian) {{MR|}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , '''1''' , Acad. Press (1976) {{MR|0460672}} {{MR|0419791}} {{MR|0412442}} {{MR|0442324}} {{ZBL|0419.31001}} {{ZBL|0339.31003}} {{ZBL|0328.33011}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> E.P. Dolzhenko, "Elimination of singularities of analytic functions" ''Uspekhi Mat. Nauk'' , '''18''' : 4 (1963) pp. 135–142 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> L.J. Riihentaus, "Removable singularities of analytic functions of several complex variables" ''Math. Z.'' , '''158''' (1978) pp. 45–54 {{MR|0484286}} {{ZBL|0351.32014}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Zoretti, "Leçons sur la prolongement analytique" , Gauthier-Villars (1911)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.V. Ahlfors, "Bounded analytic functions" ''Duke Math. J.'' , '''14''' : 1 (1947) pp. 1–11 {{MR|0021108}} {{ZBL|0030.03001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> K. Nohiro, "Cluster sets" , Springer (1960)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S.Ya. Khavinson, "Analytic functions of bounded type" ''Itogi Nauk. Mat. Anal. 1963'' (1965) pp. 5–80 (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967) {{MR|0225986}} {{ZBL|0189.10903}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0375.30014}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''2''' , Moscow (1985) (In Russian) {{MR|}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , '''1''' , Acad. Press (1976) {{MR|0460672}} {{MR|0419791}} {{MR|0412442}} {{MR|0442324}} {{ZBL|0419.31001}} {{ZBL|0339.31003}} {{ZBL|0328.33011}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> E.P. Dolzhenko, "Elimination of singularities of analytic functions" ''Uspekhi Mat. Nauk'' , '''18''' : 4 (1963) pp. 135–142 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> L.J. Riihentaus, "Removable singularities of analytic functions of several complex variables" ''Math. Z.'' , '''158''' (1978) pp. 45–54 {{MR|0484286}} {{ZBL|0351.32014}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Line 21: Line 83:
 
See also [[Analytic continuation|Analytic continuation]]; [[Analytic set|Analytic set]]; [[Removable singular point|Removable singular point]].
 
See also [[Analytic continuation|Analytic continuation]]; [[Analytic set|Analytic set]]; [[Removable singular point|Removable singular point]].
  
For quite general continuation results see [[#References|[a2]]]; Riemann's theorem has an analogue in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123054.png" />: Bounded analytic functions extend analytically across subvarieties of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123055.png" />, while all analytic functions can be analytically continued across subvarieties of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081230/r08123056.png" />. See [[#References|[a1]]].
+
For quite general continuation results see [[#References|[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 [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Section G {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Harvey, J. Polking, "Removable singularities of solutions of linear partial differential equations" ''Acta Math.'' , '''125''' (1970) pp. 39–55 {{MR|0279461}} {{ZBL|0214.10001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.B. Garnett, "Analytic capacity and measure" , ''Lect. notes in math.'' , '''297''' , Springer (1972) {{MR|0454006}} {{ZBL|0253.30014}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) {{MR|1111477}} {{ZBL|0683.32002}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Section G {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Harvey, J. Polking, "Removable singularities of solutions of linear partial differential equations" ''Acta Math.'' , '''125''' (1970) pp. 39–55 {{MR|0279461}} {{ZBL|0214.10001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.B. Garnett, "Analytic capacity and measure" , ''Lect. notes in math.'' , '''297''' , Springer (1972) {{MR|0454006}} {{ZBL|0253.30014}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) {{MR|1111477}} {{ZBL|0683.32002}} </TD></TR></table>

Latest revision as of 08:10, 6 June 2020


$ 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
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