Colloquium publications.

THE BOSTON COLLOQUIUM.

continuous along the path of integration and when also I(u) is
analytic in t = zx for all values of z upon the path of integration
and for values of x in some specified region of the x-plane. If,
as we suppose, the path is rectilinear, the values of x to be excluded are evidently those which lie on the prolongations of the
vectors from the origin to the singular points of F(x). The
region of convergence of (16) is consequently a star, whose boundary consists of prolongation of these vectors.* Thus Hadamard's
integral, when applied to the analytic continuation of a function,
is superior to Borel's in the extent of its "region of summabilitv."
This is illustrated in Le Roy's thesis t with the very familiar series:
E 1.3. (2n- 1)
n=1 2.4-....2 A
Here the coefficient of x" is
1,.. dz
rJo t//(1 z)
so that
f( I' if z
7r Jo 1z(1 -) (1 - zx)
Since F(zx) 1/(1 - zx), the region of summability is the entire
plane of z with the exception of the part of the real axis between
x = 1 and x == o. Borel's polygon of summability for the series,
on the other hand, is only the half plane lying to the left of a
perpendicular to the real axis through the point x = 1.
Much, it seems to me, can yet be done in following up the use
of Hadamard's integral. One special case has been studied already
by Le Roy, in which the (n + l)th coefficient of (I) has the form
a,= (zn()dz.
* This conclusion also holds if onlyf V(z)dz is an absolutely convergent integral, as is shown by Hadamard.
tp.411.