Note on the Picard theorem in the case of several variables

Abstract

As is well known, the Picard theorem is one of the most famous results in the theory of value-distribution of analytic functions near an isolated essential singularity. For the case of functions of several complex variables, analogous results are obtained in several ways. See W. Rothstein [4a] and K. Stein [5a].1) In the present note, the author will give a proof to the Picard theorem in the case of several variables. For convenience, we take the space of (n+1) complex variables zt, ...,zn, w, where n•†1. First we consider the case where the function f(z1,..., zn, w) is holo morphic outside a set V on which f has the essential singularities, i. e., at every