Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 95

will represent an analytic function of x over any closed region of
the x-plane which excludes the positive real axis. If, now, t
passes through any indefinitely increasing set of values,
tl < t2, < t3
we have in
ti e-zdz
a series of analytic functions which is seen at once to converge
uniformly over the region considered, since
If (A) -fi() I zXj ^ <
for sufficiently great values of i and j. The limit (2) is therefore
analytic.
By deforming the path of integration the same conclusion concerning; the analytic character of
the function (2) can be extended -o. >
to all values of x upon the positive real axis excepting 0 and x, and when the deformation is
made on opposite sides of a fixed point x, the two values of the
integral will be found to differ by
1 _1
(3) 2ir - e X.
The integral accordingly represents a multiple-valued function
with the singular points 0 and oo, the various branches of which
differ from one another by multiples of the period (3). For the
initial branch which was given in (2) the limit of f(n)(x)ln! will
be the (n + 1)th coefficient of (1) if x approaches the origin
along any rectilinear path except the positive real axis.
Let the process which has been adopted for the series of Laguerre be applied next to any other series

(I)

a. + adx + a2x2 + *..