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

180 Problems of Chapter 2
this curve under the mapping and let be chosen. If
the variation of is equal to then the variation of is
equal to (cf., 289). Consequently One may
choose for the branch for the branch and
for the branch
a)
Answer.
b)
Answer.
c)
Answer.
d)
Answer.
e)
Answer.
292. As for the function one proves that after making the
cut from the point to infinity, for example along the negative side
of the real axis, the function turns out to be decomposed into three
single-valued continuous branches. During a simple counterclockwise turn
around the point varies by during a double turn by
and only after a triple turn around the point does the value
of the function come back to its initial value. The scheme of the
Riemann surface of the function has thus the form shown in Figure
69. The Riemann surface is represented in Figure 70 2 .
FIGURE 69
293. Suppose first that the curve C does not pass through the point
Let be a function which describes the continuous variation of
(cf., Theorem 6, §2.7), and let Thus and
2 To know the meaning of the details of this figure see the section: Drawings of
Riemann surfaces.