Colloquium publications.

15 4

THE MADISON COLLOQUIUM.

~ 10. THE REPRESENTATION OF CERTAIN MEROMORPHIC
FUNCTIONS AS QUOTIENTS
In his noted memoir of 1876 Weierstrass showed that any
function of a single complex variable, which has no other singularities than poles in the finite region of the plane, can be expressed as the quotient of two integral (rational or transcendental)
functions.
The theorem was later extended by Mittag-Leffler to the case
of an arbitrary region. A function meromorphic in such a region
can be expressed as the quotient of two functions each analytic
in the region. In both cases, the numerator function and the
denominator function never vanish at the same point of the
region.
Furthermore, the region may be any continuum whatever,
and both the zeros and the poles may be chosen arbitrarily in
it. There will always exist a function with the given zeros and
poles and otherwise analytic and different from zero in the given
region.
The first of these theorems admits generalization for a function
of several variables. If f(zl,.., n,) is meromorphic at every
point of finite space, then there exist two (rational or transcendental) integral functions G(zi,.., zn) and H(z1, *.., n,) such
that
H(Zi, ***,,n)
y(zo. * =* n G(zi, * * *,Z)
Moreover, at any point at which G and H both vanish, the
representation is a normal one; i. e., G and H have no common
factor in this point; IV, ~ 1.
This theorem was stated by Poincare for the case of two variables, and he gave a proof based on harmonic functions in fourdimensional space.* A more elementary proof, which applies,
moreover, to the general case of n variables, was later published
by Cousin. t
* Acta Math., 2 (1883), p. 97.
Acta Math., 19 (1895), p. 1.