Behnke-Stein theorem

The union of domains of holomorphy , for which for all , is also a domain of holomorphy. The Behnke–Stein theorem is valid not only in the complex Euclidean space , but also on any Stein manifold. If the sequence is not monotone increasing by imbedding, the theorem is not valid; e.g. the union of the two domains of holomorphy

and

in is not a domain of holomorphy.

References

[1]	H. Behnke, K. Stein, "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität" Math. Ann. , 116 (1938) pp. 204–216
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)