Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 117

j=M i=n i /
or the remainder after the (m - l)th power of x in
(16) Z P(a)xi - cf(x) - c2f2(x) -... - c^f().
j=o
Now the series (15) before its rearrangement was a uniformly
convergent series of analytic functions and defined a function
which was analytic within (r'). It follows that (16) is also
analytic within this circle, and hence
Z P(a n)x
n=l
has no singularities within this circle except those of
fA(), f2(), '. f.-l(X).
But the radius of (r') was any quantity short of r", and this conclusion therefore holds within a circle having its center in the
origin and a radius equal to r71. By increasing n indefinitely, the
theorem of Desaint results. It is evident also that if fl(x), and
therefore f(;x), represents a one-valued function, SP(aj)c" must
also be such a function.
There remains yet for consideration the third class of cases in
which the radius of convergence of the fundamental series is 1.
If upon the circle of convergence there is any singular point with
an incommensurable argument, the continued multiplication of its
affix by itself gives a set of points everywhere dense upon the
circle of convergence. It is therefore to be expected that this
circle will be, in general, a natural boundary for IP(aj)x, and
accordingly the cases which will be of chief interest will be those
in which all the singular points upon the circle have commensurable arguments. A simple case of this character is obtained
when either (10) or (11) is chosen as the generating series. If
the former be selected, the resulting series has the form SP(n)x.
This has a special interest inasmuch as its study has proved to