Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

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which every period (P) can be expressed linearly with integral
coefficients:
(P) = m'(P') + m"(P") + - + m(k)(P(k)).
Such a set of periods is called a primitive scheme, or set, of periods.*
A periodic function which is a constant or which depends on
fewer than n arguments will evidently not come under this
definition. This will also be the case if, on making a suitable
non-singular linear transformation of the arguments, f(zi,., zn)
goes over into a function of fewer than n arguments. All other
periodic functions do come under this definition, the functions
excluded being precisely those which admit infinitely small
periods.
It is a theorem due to Riemannt that a k-fold periodic function
of p-independent variables cannot existt when k > 2p. On the
other hand, the Abelian functions have led to 2p-fold periodic
functions of p complex arguments, and such functions can also be
formed by means of quotients of theta functions of p arguments.
Theta Functions with Several Arguments.-The fundamental
theta function of a single argument~ can be defined by a series
as follows:
= (u) = i(u, a) = C E ean+2nu C + 0,
n=-00
where
a = r + si
and
r = T(a) < 0.
* I avoid the term primitive system of periods because of the confusion which
would thus be introduced, due to the other sense, above mentioned, in which
the words system of periods are used.
t Journ. fur Math., 71 (1859), p. 197 =Werke, 1 ed., p. 276; 2d ed., p. 294.
Cf. also Weierstrass, Berliner Monatsber., 1876, p. 680 =Werke, 2, p. 55.
t The maximum number of periods which an integral function can have is p.
Hermite, in Lacroix's Calcul differentiel et calcul integral, vol. 2, 6th ed., 1862,
p. 390.
~ This function appears in Fourier's Theorie analytique de la chaleur, 1822,
p. 333. It is usually thought of as due to Jacobi, who was the first to recognize
its importance in the theory of the elliptic functions; Fundamenta nova, 1829,
=Werke, 1, p. 228.