Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 83

This failure is on many accounts an unfortunate one. If a
further development of Poincare's theory is to be made - and this
seems to me both a possibility and a desirability -his definition
probably should be restricted by requiring (a) that the function
corresponding to the series shall be unlimitedly differentiable in
some interval terminating in the origin, and (b) that the derivatives of the function should correspond asymptotically to the
derivatives of the power series. These demands are satisfied in
the case of an analytic function defined by a convergent series and
seem to be indispensable for an adequate theory of divergent
series.*
Thus far we have considered asymptotic representation only for
a single mode of approach to the origin. Suppose now that an
analytic function of a complex variable x is represented by (1) for
all modes of approach to the origin, and let a0 be the value assigned
to the function at this point. Then if the function is one-valued
and analytic about the origin, it must also be analytic at this point
since it remains finite. Hence the series must be convergent.
The case which has an interest therefore is that in which the
asymptotic representation is limited to a sector terminating in the
origin. Suppose then that (1) is a given divergent series, and let
a function be sought which fulfills the following conditions: (a)
the function shall be analytic within the given sector for values of
* These requirements are formulated from a mathematical standpoint with a
view to extending the theory of analytic functions, and doubtless will be too
stringent for various astronomical investigations. Prof. E. W. Brown suggests
that for such investigations the conditions might perhaps be advantageously
modified by making the requirements for only m derivatives, m being a number
which varies with x and increases indefinitely upon approach to the critical point.
He also points out the difficulties of an extension in the case of numerous astronomical series which have the formf (x, t) = a +- alx + ax2 +- * *, where ai is a
function of x and t, af at being a convergent series. Poincar6's definition is however still applicable.
Oftentimes in celestial mechanics the only information concerning the function sought is afforded in the approximation given by the asymptotic series. An
objection to Poincare's definition is that it presupposes a knowledge of the function sought, for example, that lim f(x) -= a, when x= 0. As a matter of fact
the properties are often unknown. See in this connection p. 89 of these lectures.