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

Solutions 191
316. a) Let and be the single-valued continuous
branches of the function The scheme of the Riemann surface of the
function built by the formal method, is shown in Figure
94. The branches and
coincide. To obtain the correct scheme of the Riemann surface of the
function we therefore have to identify the branches
and This scheme is shown in Figure 95.
FIGURE 94 FIGURE 95
b) Let and be the single-valued continuous
branches of the function Thus and
Consequently is one of the single-valued continuous branches of
the function The branches of this function are:
The scheme of the
Riemann surface of the function built by the formal
method, is shown in Figure 96. The correct scheme (Figure 97) is obtained
by identifying the coincident branches and
FIGURE 96 FIGURE 97
c) Let be one of the single-valued continuous branches of the