Difference between revisions of "Tube domain"
From Encyclopedia of Mathematics
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''tube'' | ''tube'' | ||
| − | A domain | + | 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 | + | 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|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==== | ====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> | ||
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====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=15666
Tube domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tube_domain&oldid=15666
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article