# 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)