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.

4 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION
between the point thus met and infinity. On making this vector describe one complete
revolution round a, a Star-figure (as defined above) is obtained.
This star being given in a unique manner as soon as the elements (1) are assigned, I
call it the Star belonging to these elements, and I denote* it by A. In defining the star,
straight lines have been used as vectors: it is easy to see that curved lines, suitably
defined, might have been chosen for the purpose.
- In accordance with the phrase the star belonging to the elernents (1), I speak of the
function F (x), as well as of the functional branch FA (x), belonging to these elements.
These preliminaries being settled, my main theorem is as follows:Denote by A the star belonging to the elements
F(a), Fl)(a), F(2)(a).....
and by FA (x) the corresponding functional branch belonging to the same elements; let X
be any finite domain within A; and let o- denote a positive quantity as srall as we
please. Then it is always possible to find an integer n such that the modulus of the
difference between FA(x) and the polynomial
gn (=x) = /c(l)Fv(6a) (x - a)v
v
for values of n greater than Tn, is less than o- for all the values of x belonging to X. The
coefficients c() are chosen a priori and are absolutely independent of a, of F(a), F() (a),
F(2)(a),..., and of x.
It is very important to observe that the explicit knowledge of the star is not
necessary for the construction of the function gn (x). When the elements F (a), F() (a),
F(2)(a),... are once given, the star belonging to them is definite; but it does not enter
explicitly into the expression gn(x). The case is precisely the same as for Taylor's
series where the radius of the circle of convergence does not enter explicitly into the
expression.
The theorem can be proved by very elementary considerations, using especially the
fundamental theorem established by Weierstrass in his memoir Zur Theorie der Potenzreihen,
datedt 1841.
Passing from the same theorem for functions of several variables, we can easily obtain
a generalisation of my main theorem which includes the case of any finite number of independent variables.
The coefficients denoted by c() are given a priori. They are quite independent of
the special function to be represented just as are the coefficients in Taylor's series.
But the choice of these coefficients is not unique. On the contrary it can be made in
an infinitude of ways; and when conditions are given, the problem arises of making a
choice which is the best adapted to these conditions.
* As the first letter of the word da-rjp. + Ges. ~Werke, Bd. I, p. 67.