A Theorem Concerning Analytic Continuation for Functions of Several Complex Variables

Abstract

The purpose of this note is to prove the following remarkable theorem pertaining to analytic functions of several complex variables. THEOREM. Let T be a bounded open set with connected boundary C, in the 2n-dimensional (real) Euclidean space of the n complex variables x1, Xn , (n > 1). Let f(x1, , Xn) f(x) be a single-valued function which is analytic [meromorphic] in some region containing C. Then it is possible to extend f, by analytic continuation, to a function which is single-valued and analytic (meromorphic] throughout T + C (that is, in some open set containing T + C). The bracketed words afford an alternative reading of the theorem as a statement about meromorphic functions.2 The theorem, when f is analytic, is due to Hartogs; for meromorphic functions it was enunciated by E. E. Levi.3 The following proof grew out of a study of the demonstrations of the theorem given by Osgood and by A. B. Brown.4 The method is materially different from that of either of these men, however. It is the belief of the author that the proof to be given, rather long though it is in detail, is fundamentally extremely simple and easy to grasp intuitively. The detail seems necessary to avoid unjustifiably hasty conclusions.