# Tube domain

*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