Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 109

1
zx- =a, a =.
The points x = 1/3.i lie within the circle (l/R') which is the inner
circumference of the ring, while the points x = di/z before the
variation of z lie without the outer circumference (R/p). For
simplicity of presentation it may be convenient to assume at first
that these points form an aggregate
of isolated points. Suppose then R,)
that z follows any path in its plane % /z, ('/R
emerging from the circle (p). Then
the points aJz describe certain cor-\
responding paths which we will \
mark in the x-plane. At the same
time the contour c may be deformed -/z, -
continuously so as to recede before
the points a</z without sweeping
over any point 1//l3, provided merely that a/z never collides with
a point 1/i3; that is, z must never pass through a point ai/e.
Now when z is held fixed, a deformation in the contour c, subject
of course to the condition indicated, produces no change in the value
of the integral f(z), since the integrand is holomorphic between
the initial and deformed paths. On the other hand, when the
path is kept fixed and z is varied, we have the analytic continuation of f(z) in accordance with the theorem of Vaclle Poussiv.
By the two changes togetherf(z) may be continued over the entire
plane of z with the exception of the points a,3j =y 7j. To these
should, of course, be added z _= o, also z = 0 as a possible singular
point for any branch off(z) except the initial branch.
It should be observed that 7j is shown to be a potential rather
than an actual singular point. When, however, it is such a point,
the character of the point depends in general solely upon the nature
of the singularities ac and,j for (1) and (2) respectively. This fact
was noticed by Borel and demonstrated in the following manner.
Let

Co + c x + C2X2 +.*