Continuity theorem

continuity principle

Let be a domain of holomorphy in , , and let and , be two sequences of sets, with compact closures in , for which the maximum modulus principle holds for functions that are holomorphic in , that is,

Then if converges to some bounded set and to a set , and if and has compact closure in , then has compact closure in . If for one takes analytic hypersurfaces and for their boundaries , one obtains the Behnke–Sommer theorem (see [1]). Hence it follows that every domain of holomorphy is pseudo-convex. Applied to a specific function, certain modifications of the continuity theorem are known as theorems on "analytic discs" . For example, the strong theorem on analytic "discs" asserts the following. Suppose that in a Jordan curve of the form

is given. Let , , be a family of domains in the -plane having the property that for any compact set there is a number such that for all . If is holomorphic at the points of the "discs"

and at one point of the limiting "disc"

then is holomorphic also at all points of the limiting "disc" . Theorems on "analytic discs" are very useful in the holomorphic extension of domains and in constructing envelopes of holomorphy (cf. Holomorphic envelope), for example, in the proof of Bochner's theorem on the envelope of holomorphy of a tube domain, of the Osgood–Brown theorem, and of the theorem on "imbedded edges" , "the edge-of-the-wedge" , "C-convex hulls" , and others. The continuity principles given go back to the Hartogs theorem on removable singularities (1916) for holomorphic functions of several complex variables.

References

[1]	H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) (Elraged & Revised Edition. Original: 1934)
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[3]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001

Comments

The continuity principle is also known as Hartogs' Kontinuitätssatz (Hartogs' continuity theorem).

References

[a1]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 3 MR0635928 Zbl 0471.32008
[a2]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 2 MR0847923