Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 99

P. As P moves outward from the origin along any ray, the limiting position for the circle is one in which it first passes through
a singular point S, and at this point SP and OS subtend a right
angle. The region of absolute summability can therefore be
obtained as follows: Mark on each ray from the origin the
nearest singular point of the function defined by (I), if there
is such a point in the finite plane. Then through this point
draw a perpendicular to the line. Some or all of these perpendiculars will bound a polygon, the interior of which contains the origin and is not penetrated by any one of the perpendiculars. This region is called the polygon of summability. If
the singularities of the function are a set of isolated points, the
polygon will be rectilinear. For the extreme case in which the
circle of convergence is a natural boundary, the polygon and
circle coincide. In every other case the circle is included in the
polygon. Thus by the use of (4) Borel effects an analytic continuation of the series over a perfectly definite region whenever an
analytic continuation exists. On passing to the exterior of the
polygon the series ceases to be absolutely summable. As an
example of this result, take the series
x3 x5
X3 y5
+ 3- + -+...,
which is the familiar expansion of I log (1 + x)/(1 - x). The
singular points of the function are + 1 and - 1, the circle of
convergence is the unit circle, and the polygon of summability
is a strip of the plane included between two perpendiculars to the
real axis through the points -~ 1.
When the given series is divergent, the form of the domain of
summability has not been determined with such precision. The
only information which we have upon the subject is contained in
a brief but important communication by Phragmen in the Comptes Rendus,* published since the appearance of Borel's work.
Phragmen considers here the domain, not of absolute, but of simple summability for Laplace's integral
* Vol. 132, p. 1396; June, 1901.