Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

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In addition, Cousin establishes the general existence theorem
for this case, namely, that the zeros and the singularities may be
chosen at pleasure. More precisely, this condition is as follows.
To each point (a) = (al, * * *, a,) of finite space shall be assigned
a definite region T(a) including this point in its interior, and a
function
J(a)^i, ( * Zn) - (a)-(Zl -, Z,
H(a)(Zl, *, Zn)... z~) = G~a)(Zl,., z,) '
where G(a) and H(a) are both analytic in T(a) and where, in case
both functions vanish at the same point of T(a), they have no
common factor there.* When two regions T(a) and T(b) overlap,
the corresponding functions f(a) (Z, * *, Zn) and f(b) (z, * * *, )
shall be equivalent in the common region, i. e., their quotient,
taken either way, shall remain finite, and so shall have at most
removable singularities there.
Under these hypotheses there exist two integral functions,
G(Z1,, z), H(zi, *., z*), such that their quotient
H(zl, ***, Zn)
f(Z,..., zn) = G(z, Z., n)
is equivalent to f(a)(zx,. *, Zn) in the region T(a) for all values
of (a) and that, at all points at which G vanishes, this quotient
is in normal form.
From the theorems of the next paragraph it appears that both
numerator and denominator can be written as the (finite or
infinite) product of prime factors.
But Cousin's methods extend far beyond the scope of this case.
Cousin states the following theorem.t Let S= (S1,.*, Sn)
be an arbitrary cylindrical region. To each interior point
(a) = (al, *., a,) let a region T(a) lying in S and including (a)
in its interior, and a function f(a)(Zl, -*-, z,) analytic in (a)
be given. When two regions T(a) and T(b) overlap, the corre* Cf. IV, ~ 1. The denominator function G(a)(zl, * *, z,) will, of course,
in general not vanish at all, and in that case can be set = 1.
t L. c., p. 60, Theorems XIII, XIV.
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