Colloquium publications.

118

THE BOSTON COLLOQUIUM.

be of profit both for the theory of analytic continuation and of
divergent series. The reason becomes apparent when the statement is made that it is possible to throw any Taylor's series,
aan", whether convergent or divergent, into the particular form;P(n)x', and in an infinite number of ways. This fact follows
as a corollary of a very general theorem of Mittag-Leffler,* which,
when restricted to the special case before us, establishes the existence of a function P(x), which is holomorphic over the entire
finite plane and assumes the pre-assigned values a0, al, a,.. in
the points x = 0, 1, 2,.... Consequently the character of the
function defined by SP(n)xe is made to depend upon the behavior
of P(x) as x approaches oo.
Inasmuch as YP(n)ax is perfectly general, limitations must be
imposed upon P(u) in any attempt to extend Hadamard's theorem
to this series. But whenever the theorem is applicable, the only
possible singularities of;P(n)x7 are x = 0, 1, oo. Lean t establishes the correctness of this result when P(u) is an entire function
of order less than 1,1 giving also a more general theorem ~ concerning YP(a,)x:n of which this is a special case. The like conclusion
holds concerning the singular points of.P(1/n)xn, provided only
that P(x) is holomorphic at the origin. l
Very recently these results of Leau have been proved more
simply by Faber, but in a more restricted form, an artificial cut
being drawn from x = 1 to x = oo to obtain a one valued function. In addition, Faber shows that if for any prescribed e and
for a sufficiently large r the inequality
(17) 1 P(rei) I< ee
*Acta Math., vol. 4 (1884), p. 53, theorem D. For a reference to this theorem
I am indebted to Professor Osgood. Theorem 2 of Desaint's memoir (p. 438)
is in contradiction with this, but his proof is here inadequate since rk (p. 440) has
not necessarily a lower limit.
tLoc. cit., p. 418.
t He also shows that 2P(n)xn is then a one-valued function.
~Loc. cit., p. 417. See also Bull. de la Soc. Math. de France, vol. 26 (1898),
p. 267.
I[Loc. cit., p. 418; see also p. 407.