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.

2 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION
ca analytical representation of c branch FK(x) which is to be chosen as extensive as
possible.
Merely from the definition of the analytic function F(x) and from that of the branch
FK (x), there follows at once a kind of analytical representation of the branch FK (x)
in question. In effect, such a representation is always given by an enumerable number
of analytical continuations of P (xa). But as the radius of the circle of convergence
of such an analytical continuation is given only by Cauchy's criterion already quoted,
this mode of representing FK (x) becomes extremely complicated and rather unworkable.
The analytical continuation ought rather to be regarded as the definition of the function
than as a mode of representation.
There is another mode of representation which arises immediately from the principles
upon which Cauchy's theory of functions is based. Such a representation is given by
the formula
K ()= K ) dz.................................... (3),
where the integral is taken along a closed contour S within K. By the definition of an
integral, it is clear that the integral (3) can be replaced by an infinite sum of rational
functions of x, the coefficients of which are expressed by special values of x (there being
an enumerable number of these) and the corresponding values of FK1(x). This observation
was the point of departure of the investigation of M. Runge* as well as of the subsequent
investigations of MM. Painlevé, Hilbert and others. The analytical representation thus
obtained accordingly requires a knowledge of the value of FK (x) at an infinite and
enumerable number of points. Now in the customary problems of analysis these values
are not known. In general it is, on the contrary, the series of values
F(a), F( (a), F2 (a),...
which is given. Adopting the usual point of view, it is thus for instance in the problem
of the integration of differential equations.
When, then, we have to find the analytical representation of FK (x), it must be drawn
from the elements (1) and, by means of those elements alone, a formula must be constructed
to represent the branch FK(x) completely. Let C denote the circle of convergence of the
series (2). The expression
e 1! F ) (a) (x - ac)
1
then gives the analytical representation of FC(x), the equality
FC (x) - 1 F) (a) (x - a)
holding for all points within C. This expression is constructed by means of the elements
F (a), F() (a), F' (a),....
* " Zur Theorie der eindeutigen analytischen Functionen," ~ 1, pp. 229-239, Acta Mathematica, tome 6.