Colloquium publications.

76

THE BOSTON COLLOQUIUM.

circle of convergence is not a natural boundary. Then the series
defines within this circle an analytic function. In the region of
divergence without the circle the value of the function may be
obtained by the familiar process of analytic continuation. Theoretically the determination of the function is a satisfactory one,
for Poincare * has shown that the function throughout the domain
in which it is regular can be obtained by means of an enumerable set of elements, P(x - an). Practically, however, when
Weierstrass' process is employed for analytic continuation, the
labor is so excessive as to render the process nearly valueless
except for purposes of definition. Hence to-day a search is being
made for a workable substitute. I may refer particularly in this
connection to the investigations by Borel and Mittag-Le/Jler. As
I consider the work of the former to be both suggestive and
practical, I have taken it as the basis of my second lecture.
A second aspect of our topic, intimately connected with the
continuation of the function defined by (1), is the determination
of the position and character of its singularities in the region
where the series diverges. This subject is treated in Lecture 3.
When the circle of convergence is a natural boundary, it does
not appear to be impossible, despite the earlier view of Poincare
to the contrary,t to discover, at least in a certain class of cases,
an appropriate, although a non-analytic mode of continuing the
function across the boundary into other regions where it will be
again analytic. The thesis of Boreland its recent continuation in the
Acta JMathematica, together with some excellent remarks by Fabry,.
appear to be about all that has been done in this direction. A very
brief discussion of the subject will be given in the fourth lecture
in connection with series of polynomials and of rational fractions.
Lastly, we have the conundrum of the truly divergent power
series - the series which converges only when x = 0. It is upon
* Rendiconti del COrcolo Matematico di Palermo, vol. 2 (1888), p. 197, or see
Borel's Theorie des fonctions, p. 53.
tThe conclusions of Poincare and Borel are not actually inconsistent, but a
new point of view is taken by the latter.
+ Compt. Rend., vol. 128 (1899), p. 78.