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.

8 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION
stars C(I), C(2), C),... each of which is circumscribed to that which precedes it and is
inscribed* to that which follows it; to these there correspond expansions PC() ( la),
PC(2) (x a), PC() (x a),... which preserve all the principal characters of the Taylor's series
PC(x a). The expression PC() (x a) is merely a (/ + l)-ple series with limited convergence.
There is another method of generalising Taylor's series as follows:
Denote by A a star with its centre at a, and by A () an associate star, concentric
with A and inscribed in A, defined with reference to A in some suitable manner. This
star A(') is to be such that it becomes a circle when S = 1 and that it encloses in its
interior every domain within A when the quantity S is sufficiently small.
Now suppose that A is the star belonging to the elements F(a), F(W> (a), F(> (a),...,
and construct the series
P (l a)=F(a)
+ Z {h, (> (<) F(' (a) (x - a) + h,>() (S) F (2) () (x - a)2 +... + h() () F() (a) (a) - a)}..(5).
À=l
The coefficients
h(À) (:),')
( =1, 2,..., X)'
can be assigned a priori, independently of a, of F(a), F(>) (a), F(2) (a),... and of x, so
that the series possesses the following properties: it converges for every point within A (8
and converges uniformly for every domain within A( ). If convergence takes place for any
value, the value necessarily belongs to the interior of A(8) or is a point of the star A( ).
When 8=1, the series becomes Taylor's series.
The equality
FA (x) = P3 (x1 a),
exists throughout the interior of A ().
Among other differences between the two generalisations of Taylor's theorem, this
may be noted: that in the first the stars 0(l), C(2), C(3>,... form, so to speak, a discontinuons
sequence of domains of convergence, while in the second there is a continuous transition
from the circle C (== A)) to the star A (= A(~).
The star which belongs to the elements F (a), F('> (a),... is given at the same time
as these elements, just as the circle which belongs to the elements also is given. But
in order actually to construct the star on the circle, we must in the first case know
the points of the star (it is thus that I describe the points formerly denoted by a,)
and in the second case the distance between a and the nearest point of the star.
It might be difficult to deduce this knowiedge simply by the study of the elements
F(a), F(') (a), F(2) (a),.... But in some problems the points of the star are directly given:
e.g. the determination of the general integral of a differential all of whose critical points
are fixed, being finite in number. In this case, we can construct the star directly and
can obtain an analytical expression for the integral valid over the whole plane except
* A star is inscribed in another which circumscribes it if the whole of the first star belongs to the second and if
the two stars have common points such as a2.