Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

145

n - 1 north poles combined with any point whatever of the
nth sphere, then f is a constant.
As a special case of the theorem we have the
Corollary. If f(zi, * *, zn) is analytic in every point of the
infinite region of the space of analysis, then f is a constant.
This last result can be stated in a form wholly independent of
any assumption regarding the infinite region.
Theorem B. If f(zi, -* *, zn) is analytic at all finite points outside a fixed hypersphere:*
G < xI2 + y12 + 22 + * + yn2,
and if f is finite in this region, then f is a constant.
Returning now to theorems, relating to rational functions,
we have the following.
Theorem 2. If f(zi, ***, zn) is a rational function of each
individual variable, when all the others are assigned arbitrary
values in the neighborhood of a certain fixed point and then held
fast, then f is rational in all its arguments.
The proof of Theorem I is covered by Hurwitz's reasoning, and
the same is true of Theorem II, provided the additional hypothesis is made that the function be analytic in all its arguments
in the neighborhood of the fixed point in question. In practice,
this further condition appears usually to be fulfilled. For a
proof that this condition is a consequence of the others I am
indebted to Professor E. E. Levi.
Both theorems can be extended to algebraic functions, the
hypothesis then being that the function is N-valued, and that,
moreover, it is algebroid, where before it was meromorphic.
~ 7. ON THE ASSOCIATED RADII OF CONVERGENCE OF A POWER
SERIES
Let
ECv-vX1 1 *v * Xnv.
be a power series convergent for a set of values of the arguments,
* This hypothesis may equally well be written in the form
G < izil + * + Iznl.