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.

I. On the analytical representation of a uniorm branch of a monogenic function.
By G. MITTAG-LEFFLER.
[Received 25 April, 1899.]
LET a denote a point in the plane of the complex variable x, and associate with
a an unlimited array of quantities
F), (a), F() (a), (a., F) (a),.......................... (1),
where each quantity is completely determinate when the position which it occupies in
the array is known.
Suppose that, as is possible in an infinite number of ways, these quantities F are
chosen so that Cauchy's condition*, that the series
P(l|a)= S! (i) (a) (x - a)b.......... (2),
shall have a circle of convergence, is satisfied.
In the theory of analytic functions constructed by Weierstrass, the function is defined
by the series P (x a) and by the analytic continuation of this series. The function is
completely determinate provided the elements
F(a), F(l (a), F(2) (a),..., F() (a),...
are given. We denote generally by F(x) the function in its totality which is defined
by these elements.
If K is a continuum formed by a single piece, which nowhere overlaps itself and
encloses the point a, and if it is such that the branch of the function F (x) formed
by P (x a) and by its analytic continuation within K remains uniform and regular, I
shall denote this branch by FK(x). The problem to be discussed here is that of finding
* Cauchy, Cours d'Analyse de l'École royalepolytechnique, is a finite magnitude. It is known that, if this finite
1lre partie, Analyse Algébrique, Paris 1821, chapitre 9, ~ 2, 1
theorème I, p. 286. Expressed in modem phraseology, magnitude be denoted by -, the quantity r is the radius of
theorème i, p. 286. Expressed in modern phraseology,
Cauchy's condition would be formulated thus: The upper the circle of convergence of the series (2).
limit of the limiting values of the moduli.!F( (a)XI 1
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