Colloquium publications.

144

THE MADISON COLLOQUIUM.

Corollary. If f(zi, *., Zn) is meromorphic in every point of
the infinite region of the space of analysis, then f is a rational
function of all its arguments.
These theorems readily suggest others, in which the word
meromorphic is replaced in the hypothesis by analytic; the conclusion then being that the function is a constant.
Theorem A. If f(z1, * *, Zn) is analytic in every point of the
coordinate axes, then f is a constant.
The manifold M consisting of the coordinate axes is perfect,
and hence f is analytic in a 2n-dimensional region T enclosing
the axes. It is possible, in particular, to choose a positive number
h so that f is analytic in the region
Zlli < h, *...*, |Zk-11 < h, lZk+l| < h, *...*, 1Znl < h,
Zk ranging over the whole extended zk-plane; k = 1,...* * *, n.
Consider f in the region
2: \Zkl < h, k- l,..., n.
Let (a, * * *, an) be a point of this region. The function
f(a,.* *.., an-1, zn)
is analytic over the whole extended zn-plane. Hence it is a constant. Hence
0Qf(Z..-, n)
OZn
in the point (a, * * *, an). But this was any point of 2.
It appears, then, that
= 0, k =,..., n,
and from this fact follows the truth of the theorem.
As in the case of Theorem 1, so here the theorem admits an
alternative statement.
Theorem A'. If f(z, * * *, Zn) is analytic in those points of the
infinite region of the space of analysis which correspond to any