Abel's theorem in problems and solutions - School of Mathematics

82 Chapter 2
286. Fix the value at a certain point and define the
values of function at the other points of the plane (except the cut) by
continuity along the curves starting from and not intersecting the cut.
Prove that the continuous single-valued branches so obtained coincide
with the function (defined by the value at point
It follows from the result of Problem 286 that, choosing as initial
point different points of the plane, one obtains the same splitting of the
Riemann surface into single-valued continuous branches. Therefore this
splitting depends only on the way in which we have made the cut.
287. Suppose that points and do not lie on the cut and that the
curve C, joining them, traverses the cut once (Figure 31). Choose a value
and by continuity along C define the value Prove
that values and correspond to different branches of
In this way, traversing the cut, one moves from one branch to the
other, i.e., the branches join each other exactly as we have put them
together joining the sheets (Figure 29). One obtains in this way the
Riemann surface of the function
We say that a certain property is satisfied by any turn around a point
if it is satisfied by a simple turn counterclockwise along all circles
centred on and with sufficiently small radii 14 .
288. Prove that by a turn around a point one remains on the same
sheet of the Riemann surface of the function if and one moves
onto the other sheet if
The following notion is very important in the sequel.
DEFINITION. Points, around which one may turn and move from one
sheet to another (i.e., changing the value of the function) are called the
branch points of the given multi-valued function.
The Riemann surface of the function can be represented by a
scheme (Figure 32).
This scheme shows that the Riemann surface of the function
has two sheets, that the point is a branch point, and that by turning
around the point one moves from one of the sheets to the other.
Moreover, the arrow between the two sheets in correspondence with the
point indicates the passage from one sheet to the other not only
by a turn around the point but also by crossing a point of the
14 More precisely, this means that there exists a real number such that the
property mentioned is satisfied by any turn along any circle with centre and radius
smaller than