Generalization of Levi-Oka’s Theorem Concerning Meromorphic Functions

Abstract

As Fuks [3] stated, every domain of holomorphy or meromorphy over Cn is analytically convex in the sense of Hartogs. Oka [6] proved that every domain over Cn analytically convex in the sense of Hartogs is a domain of holomorphy. Therefore a domain of meromorphy over Cn coincides with a domain of holomorphy over Cn.

Propagation of analyticity for essentially finite -smooth CR mappings

“…In summary, at every point q of Ω, a meromorphic extension of the holomorphic function A 1 (x, y)/A 2 (x, y) defined in Ω\Z A2 has been constructed. It follows from [18] that this meromorphic function can be represented as a quotient of holomorphic functions in Ω, which completes the proof of the lemma.…”

“…According to the works of K. Oka and E.E. Levi, a meromorphic function defined in a domain of C n is always the global quotient of two holomorphic functions (see [18] for instance). It follows that there exist functions R ℓ = R ℓ (t) holomorphic in V 3 such that (5.10)…”

Meromorphic or Holomorphic Completion of a Reinhardt Domain

“…Kajiwara and the author [2] have proved this result for any domain over a Stein manifold. Therefore the above result is established naturally as the special case of [2]. However, for the elementary domain like the Reinhardt domain, it is desirable to give the direct and simple proof.…”

Analytic continuation and boundary continuity of functions of several complex variables

“…Both g and h continue into D. (That the Poincare problem is solvable on arbitrary domains in C n is well known but does not seem to appear in the standard texts on several complex variables. A proof is given in [14]. The point is that the envelops of holomorphy and meromorphy of a domain in C" are the same.…”