Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 101

By the change of variable w = Oz this becomes
J 0 e-I (w)e~ dw I0/ (w)ewdw.
0 O- al0 Joal
Since e-w(t/8-) is a positive and decreasing function in the interval
considered, the second mean-value theorem of the integral calculus*
may be applied, giving
e-ai(l-0) rOa
(11) J = d_ 4)(w)e-dw,
in which a designates an appropriate value between a, and a2.
This, as Phragmen says, proves the theorem, but a word or two
of explanation additional to his " deux mots" may not be unacceptable to some of my hearers. The necessary and sufficient
condition for the existence of the first of the two integrals given
in (10) is that by taking two values a, and a2 sufficiently small or
two values sufficiently large, the integral J may be made as small
as we choose. Now this is true of
Pa
0(w)e6-'tdw
X2
since the integrals (9) exist, and equation (11) show then that it
must be true likewise of J because the factor e-a(l-0)/O has an
upper limit for 0 <0 < < 1 and 0 < a < a<. It follows
therefore that the integrals (10) exist.
Two other facts stated by Phragmen are also of interest. The
function of x defined by (8) is a monogenic function which is holomorphic at every point in the interior of a circle described upon
OP as diameter. If, also, in place of f(zx) we take the associated
series F(zx) of a convergent series (I), the star of convergence
coincides with Borel's polygon of absolute summability. Thus
the regions of absolute and non-absolute summability are the
same, or differ at most only in respect to the nature of the boundary points.
Bonnet's form: Encyklopidie der Math. Wiss., II A 2, ~ 35.