A Theorem on Analytic Continuation of Functions in Several Variables

0 xX = Xx ; _x < ye < ??, K = 1, **k where (x?, ... , x) is an arbitrary point of S. A tube is an (open) domain if and only if its basis is a domain. We shall say that a tube T' lies within a tube T if the closure of T' is part of the interior of T. The convex closure (in the usual sense) of the tube T will be denoted by T. Obviously T is again a tube and its base S is the convex closure of S. In the paper loc. cit. it was shown that if a function f(z) = f(z1, * , Zk) is analytic and bounded within T it also exists and is bounded within T. In the present paper we shall establish the theorem omitting the property of boundedness. THEOREM. Any function which is analytic in a tube T is analytic in its convex closure T. On the other hand, any convex tube T is the natural domain of analyticity for some function. In fact, there exists functions of this kind which have the period 27ri in each variable z5 . Combining this fact with our theorem we are led to the following statement: the envelope of regularity (Regularitatshiille)2 of T is T. From the viewpoint of the theory of analytic functions of real variables our theorem may be formulated as follows. A function f(x,, * * , Xk) which is analytic over a domain S has, by definition, an extension