Colloquium publications.

142

THE MADISON COLLOQUIUM.

~ 4. RATIONAL AND ALGEBRAIC FUNCTIONS
To Weierstrass is due the theorem that a function of n complex
variables which is meromorphic at every point of the space of
analysis is a rational function.*
Weierstrass did not define the space in which the function is
considered. He said "im ganzen Gebiete seiner Veranderlichen." It appears, however, from more explicit statements
in similar casest that he thought of each variable as an arbitrary
point of its extended plane.
A similar theorem holds for algebraic functions. If a function
of n complex variables is finitely multiple-valued and if, in the
neighborhood of every point of the space of analysis, the values
of the function can be so grouped as to satisfy one or more algebroid relations,
Aowm + A1Z w-i + * ' + A, = 0,
where the A's are analytic in the point in question, - and to be
exhausted in said neighborhood by these systems, -then the
function is algebraic.
~ 5. SUFFICIENT CONDITIONS THAT A FUNCTION OF SEVERAL
COMPLEX VARIABLES BE ANALYTIC
In order that a function of two real variables be analytic it is
not enough that the function be analytic in each variable separately when the other is held fast, as is shown by the example:
xy
f(x, y) = x+ y2 0< xl + lyl;
f(0, 0) = o,
the function being considered in the neighborhood of the origin.
When, however, we allow the variables to take on complex values,
the case stands otherwise.
* Journ. fiur Math., 86 (1880), p. 5=Werke 2, p. 129. The theorem was
proven by Hurwitz, Journ. fir Math., 95 (1883), p. 201.
t Cf. for example Werke, 3, p. 100, 7th line from end.