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.

6 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION
C. Between the circle and the star, intermediary domains C(0), (u= 1, 2, 3,...), exist,
unlimited in number; each of them in succession includes the domain that precedes it;
and they can be chosen so that, corresponding to each domain C(G), there is an analytical
expression representing FA (x) within C(w) which converges uniformly for every domain
within C(0) and ceases to converge outside C(G). On this question there is an interesting
study to be made which I have merely sketched in my Swedish memoirs; to it I shall
return on another occasion.
The only writer who, so far as I know, has found a general representation of FA (x)
valid outside the circle belonging to the elements (1) is M. Borel. In two important
memoirs*, M. Borel is concerned with what he calls the summability of a series. It appears
to me that the chief interest of this investigation of M. Borel is that the author really
finds an expression valid for a domain which in general includes the circle C. The
domains which I have called C(G) can easily be chosen so that C(1) becomes this domain
K: so that M. Borel's domain K becomes the second approximation to the Star, the
circle being the first as already indicated.
But M. Borel has discussed the same class of ideas in another publication. In his
book-t published without any acquaintance with my Swedish Notes of the same year, the
author says+:"Pour résumer les résultats acquis sur le problème de la représentation analytique
"des fonctions uniformes, nous pouvons dire~ que nous en connaissons deux solutions
"complètes; l'une est fournie par le théorème de Taylor, l'autre par le théorème de
"M. Runge l. Ces deux solutions ont une très grande importance à cause de leur
"généralité; mais chacune d'elles a de graves inconvénients dont les principaux sont, pour
"la série de Taylor, de diverger en des régions où la fonction existe; et, pour la repré"sentation de M. Runge et celles de M. Painlevé, d'être possibles d'une infinité de
" manières ~..................................................................................
"Le but idéal à atteindre, c'est de trouver une représentation réunissant les avantages
"de la série de Taylor et des séries de M. Runge ou de M. Painlevé, sans avoir aucun
"de leurs inconvénients", et le but immédiat, c'est de trouver une telle représentation
"pour des classes de fonctions de plus en plus étendues +-t."
* Journal de Mathématiques, 5-e Sér., t. ii. (1896), require the knowledge of an enumerable number of values
"Fondements de la théorie des séries divergentes som- of the function which correspond to points that approach
mables," pp. 103-122; " Sur les series de Taylor admettant indefinitely near the limit of existence of the function.
leur cercle de convergence comme coupure," pp. 441-454. ** It will be seen that I have achieved this aim, not only
+ Leçons sur la théorie des fonctions, Paris, 1898. for uniform analytic functions but also for the functional
+ pp. 88 ff. branch FA (x). It might be asked whether it would not be
~ All that follows on the analytical representation of possible to achieve the same aim for the function F (x) in
uniform functions can be applied, mutatis mutandis, to the its totality. It is not so: such a question is too general.
functional branch FA (x). The problem was mainly that of limiting the question so as
I I have indicated above that, in M. Runge's theorem, to make a solution possible without diminishing the
there is nothing which is not already in principle contained generality more than was necessary. I believe that this
in the representation by Cauchy's integral. problem is solved by the introduction of the star and of the
~T In what precedes, I have pointed out what appears to functional branch FA (x).
me a graver inconvenience, viz. that these expressions ++ It appears that M. Borel has not regarded his own