Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 105

so that
(14) as= f(z) -zdz.
At first sight this choice of functions would seem to be a very desirable one, for the function defined by the divergent series is obtained
in the familiar form
(15) 4 fz)- (r 1 -
Upon examination, however, it turns out to be otherwise. For
suppose the divergent series to be given and f(z) is to be found.
The problem is then a very difficult one, that of the inversion of
the integral (14) when an is given for all values of n. This is
what Stieltjes terms " the problem of the moments." It does not
admit of a unique solution, for Stieltjes himself * gives a function,
f(z)- e- sin /z,
which will make ask= 0 for all values of n. If the supplementary
condition is imposed that f(z) shall not be negative between the
limits of integration, only a single solution f(z) is possible, but
the divergent series is thereby restricted to belong to that class
which Stieltjes derives naturally and elegantly by the consideration of his continued fraction.
Thus far our attention has been confined exclusively to integrals
in which one of the limits of integration is infinite. There are,
however, advantages in using appropriate integrals having both
limits finite, at least if the given series is convergent and the
integral is used for the purpose of analytic continuation. In
particular, the integral
(16) f(x) V(z)F(zx)dz
should be noted, to which Hadamard has drawn attention in his
thesis.t This falls under Tallee-Poiessin's theorem when T(z) is
*Loc. cit., ~ 55.
tJourn. de Math., ser. 4, vol. 8 (1892), pp. 158-160.