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.

OF A LTNIFORM BRANCH OF A MONOGENIC FUNCTION. 3
and of the rational numbers independent of the choice of the elements: and it is to
be remarked that the expression is formed without any a priori knowledge of the radius
of the circle C. This radius is determinate, in connection with the elements, by Cauchy's
theorem, and there are various rnethods of obtaining it from them; but it does not enter
explicitly into the expression. Thus Taylor's series is formed simply by the elements
F (a), F(1) (a), F(2' (a),...,
when these are the derivatives of the function.
The following question may therefore be proposed: Is it possible to obtain for a branch
FK(x) with the greatest range possible an analytical representation of this nature? As I
have shewn in various notes, published in Swedish by the Stockholm Academy of Sciences
during the past year, the reply is in the affirmative, and consequently it is possible to fill
an important lacuna in the theory of analytic functions. In fact, hitherto it has been
impossible to give for the general branch FK(x) an analytical representation similar to that
found from the very beginning of the theory for the branch FC(x).
For a fundamental treatment of the question which has been proposed, it is first
necessary to define a domain K which shall be as great as possible. This I shall do by
the introduction of a new geometrical conception-a Star-figure.
In the plane of the complex variable x, let an area be generated as follows. Round
a fixed point a let a vector I (a straight line terminated at a) revolve once: on each position
of the vector, determine uniquely a point, say a1, at a distance from a greater than a
given positive quantity, this quantity being the same for all positions of the vector. The
points thus determined may be at a finite or at an infinite distance from a. When the
distance between a, and a is finite, the part of the vector from a, to infinity is excluded
from the plane of the variable.
The region which remains after all these sections (coupures) in the plane of x have
been made is what I call a Star-figure. Manifestly the star is a continuum formed of a
single simply-connected area.
Associate with a the elements
F(a), F() (a), F(2) (a),..., F( (a),...
satisfying Cauchy's condition; and form the series
P (x a)= 1! F (a) (x -a)-t.
Construct the analytical continuation of P(x a) along a vector from a. It may be the
case that every point of this vector belongs to the circle of convergence of a series which
itself is an analytical continuation of P(xla) obtained by proceeding along the vector; but
it is also possible that, on proceeding along the vector, a point is met not situated within
the circle of convergence of any analytical continuation of P(xla) along the vector. In the
latter case, I exclude from the plane of the variable that part of the vector comprised
1 — 2