Note on the Picard theorem in the case of several variables

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