Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

157

write it as the product of two integral functions:
G(zl, * *, Zn) = Gl(zl, ***, zn)G2 (Z1, *., n),
both of which vanish.*
The roots of a prime function yield the coordinates of all
finite points of a certain monogenic analytic configuration.
Let G(zi, * *, Zn) vanish in a point, but not vanish identically.
Then the equation
G(z1,.*, Zn) = 0
defines one or more monogenic analytic configurations. Let M
denote one of them. By the aid of Cousin's theorem it is possible
to infer the existence of an integral function which vanishes in
the points of M and nowhere else, and which, moreover, is prime.f
G(zi,.., Zn) is divisible by this function.
From Weierstrass's factor theorem, IV, ~ 1, it now follows
that, in the neighborhood of a point and hence throughout any
finite region of 2n-dimensional space, an integral function which
vanishes there, but does not vanish identically, can be written
as the product of a finite number of factors, each irreducible in
the point or in the region, multiplied by another integral function
which does not vanish there.
It is now an easy matter, by the methods used in the proofs of
Weierstrass's and Mittag-Leffler's theorems, to establish the
proposed generalization: An integral function which vanishes,
but does not vanish identically, can be written in one, and
essentially in only one, way as the (finite or infinite) product of
its prime factors.
Moreover, the existence theorem for such functions, whose
prime factors are arbitrary, holds there. Let G1, G2, *.. be an
infinite set of prime functions subject merely to the condition
that at no point of finite space do the monogenic analytic configurations which correspond to their roots have a cluster point.
* Gronwall, Thesis, Upsala, 1898, p. 7.
t This theorem is due to Gronwall, 1. c. It was rediscovered by Hahn,
Monatshefte, 16 (1905), p. 29.