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

76 Chapter 2
FIGURE 24
valued and continuous over the whole plane except the cut. This function,
which we denote by defines a continuous single-valued mapping
of the plane except the cut, on the left half plane (Figure 25). Here
FIGURE 25
Functions and so defined are called the continuous singlevalued
branches of the function (for the given cut).
Consider now two copies of the plane, which we shall call sheets,
and cut every sheet along the negative side of the real axis from 0 to
(Figure 26).
Let us take the function on the first sheet and the function
on the second sheet. Thus we can see the functions and as a
unique single-valued function, defined no longer on the plane but on a
more complex surface consisting of two distinct sheets.
So if a point moves continuously on the first (or on the second) sheet,