Colloquium publications.

128

THE BOSTON COLLOQUIUM.

an upper limit to the degrees of Pn (x) and Rj(x). But this is not
enough, and he proceeds therefore to limit the magnitude of the coefficients in the numerators. On the other hand, he allows any distribution whatsoever for the roots of the denominators, thus leaving
himself at liberty to vary greatly the nature of the function represented.
In his thesis he develops the case
0o A
(6) cr)= ( n (m - m)
which had been previously considered by Poincare * and Goursat.t
To avoid semi-convergent series or, in other words, functions, of
which the character depends not merely upon the position of the
poles an and the values of A. but also upon the order of summation,
the condition is imposed that EAn shall be absolutely convergent.
Then if there is any area of the z plane which contains no poles,
the series (6) must converge within this region. Since furthermore it is uniformly convergent in any interior sub-region, it
defines an analytic function within the area. There may be
several such areas separated by lines or regions in which the poles
are everywhere dense. This is precisely the case to be considered
now.
To simplify matters, let us suppose that the poles are everywhere dense along certain closed curves of ordinary character,
but nowhere inside the curves. Poincarg and Goursat show that
each curve is a natural boundary for the analytic function +(z)
defined by (6) in its interior. Borel's proof is as follows. Denote the component of (6) which corresponds to a. by
~B) B_ 1 B1
and the remaining part by

* Acta Societatis Fennicce, vol. 12 (1883), p. 341, and Amer. Journ. of Math.
vol. 14 (1892), p. 201.
t Compt. Rend., vol. 94 (1882), p. 715.