Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

OF A UNIFORM BRANCH OF A MONOGENIC FUNCTION. 7
There exist a certain number of other investigations having relations with my theorems
but belonging to a range of ideas quite different from M. Borel's. I have already spoken
of the representation which follows from Cauchy's integral
FA (x) = 2 -f - - dz.
With M. Runge, we can transform this integral into a series every term of which is a polynomial in x. But in order to construct these polynomials, it is necessary to know not only
the star A but also the values of the function for an enumerable number of points
approaching indefinitely near the boundary of A. Investigations have been carried out
in which the elements F(a), F(i) (a), F(2> (a),... are substituted for these values of the
function. But these investigations always abut, in a manner more or less direct, upon
the conformal representation of the circle of convergence on another figure known beforehand: and they still require that we should know, as to the function which is to be
represented, that it is regular within the domain represented on the circle. The most
interesting and the most significant theorem in this range of ideas appears to me to
be that of M. Painlevé:
Given a convex domain D and an internal point a, a set of polynomials
- n0 (x), ni (,),..., Hn (x); (, = 1, 2, 3,...),
can be constructed such that any function F (x) holomnorphic in D is developable in that
domain in the form
F(x) = I {F, (a),, (x) + F 1' (a) HU (x) +... + F(, (a) I, (x)}.
z=O
The resemblance between M. Painlevé's formula and mine is obvious. Writing
IIuv (x) = t() (x - a)
in M. Painlevé's formula, mine follows. Yet the resemblance is entirely formal, because the
formation of the polynomials II,, (x), T,j (x),..., HI, (x) requires the a priori establishment
of the domain D and the knowledge of the function F(x) that it is holomorphic in D:
whereas with me the formula of representation, so far from supposing any a priori knowledge of the star A, gives on the contrary the means of determining the starF.
In other publications, it is my intention to develop other theorems in the same range
of ideas as well as to return to the numerous applications that can be made of my
theorems: I restrict myself in this place to the following indications. I have just
explained that, besides the circle C and the star A, there is an infinite number of other
investigations on the summability of series from the point appeared in the Comptes Rendus (23 May, 1899). In the
of view just indicated so clearlyin his book. Otherwise he same number of the Comptes Rendus, there is a note by
rather might have said: that the immediate aim was to M. Borel related to my investigations. The reader is also
find a general representation valid for a domain still more referred to an addition to the "mémoire sur les séries
extensive than this domain K (that is, C()). divergentes par E. Borel" (Ann. de l'Éc. Norm., 1899), and
* Comptes Rendus, t. cxxvi (24 Jan., 1898), pp. 320, 321. to two important notes by M. Picard (Comptes Rendus,
+ While the present note was passing through the 5 June, 1899) and M. Phragmén (Comptes Rendus,
press, a new and interesting note of M. Painlevé's, dis- 12 June, 1899): all of them are connected with these
cussing the relation of these investigations to my own, has investigations.