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''tube''
 
''tube''
  
A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944101.png" /> in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944102.png" /> of the form
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A domain $  T $
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in the complex space $  \mathbf C  ^ {n} $
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of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944103.png" /></td> </tr></table>
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$$
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= B + i \mathbf R  ^ {n}  = \
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\{ {z = x + iy } : {x \in B, | y | < \infty } \}
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,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944104.png" /> is a domain in the real subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944105.png" />, called the base of the tube domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944106.png" />. A domain of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944107.png" /> is also called a tube domain. The [[Holomorphic envelope|holomorphic envelope]] of an arbitrary tube domain is the same as its convex hull; in particular, every function that is holomorphic in a tube domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944108.png" /> can be extended to a function that is holomorphic in the convex hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t0944109.png" />. A tube domain is said to be radial if its base is a connected cone in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094410/t09441010.png" />.
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where $  B $
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is a domain in the real subspace $  \mathbf R  ^ {n} \subset  \mathbf C  ^ {n} $,  
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called the base of the tube domain $  T $.  
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A domain of the form $  \mathbf R  ^ {n} + iB $
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is also called a tube domain. The [[Holomorphic envelope|holomorphic envelope]] of an arbitrary tube domain is the same as its convex hull; in particular, every function that is holomorphic in a tube domain $  T $
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can be extended to a function that is holomorphic in the convex hull of $  T $.  
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A tube domain is said to be radial if its base is a connected cone in $  \mathbf R  ^ {n} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of many complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of many complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Hörmander,  "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Chapt. 2.4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Hörmander,  "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Chapt. 2.4</TD></TR></table>

Latest revision as of 08:26, 6 June 2020


tube

A domain $ T $ in the complex space $ \mathbf C ^ {n} $ of the form

$$ T = B + i \mathbf R ^ {n} = \ \{ {z = x + iy } : {x \in B, | y | < \infty } \} , $$

where $ B $ is a domain in the real subspace $ \mathbf R ^ {n} \subset \mathbf C ^ {n} $, called the base of the tube domain $ T $. A domain of the form $ \mathbf R ^ {n} + iB $ is also called a tube domain. The holomorphic envelope of an arbitrary tube domain is the same as its convex hull; in particular, every function that is holomorphic in a tube domain $ T $ can be extended to a function that is holomorphic in the convex hull of $ T $. A tube domain is said to be radial if its base is a connected cone in $ \mathbf R ^ {n} $.

References

[1] V.S. Vladimirov, "Methods of the theory of functions of many complex variables" , M.I.T. (1966) (Translated from Russian)

Comments

References

[a1] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4
How to Cite This Entry:
Tube domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tube_domain&oldid=49044
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article