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

184 Problems of Chapter 2
is a continuous curve. If the curve with equation is one of
the continuous images of the curve under the mapping
then the curves with the equations
are, as well, continuous images of the curve under the mapping
(cf., 296), and, consequently, they are continuous images of
the curve C under the mapping Thus if the value of the
function at the initial point is multiplied by the value
at the final point of the curve defined by continuity, will be multiplied
by Therefore if is a continuous single-valued branch of the
function then all continuous single-valued branches are obtained
by multiplying by
Answer.
303. a) During a turn around the point or
varies by (cf., 260), and varies by (cf., 280), i.e.,
the value of the function is multiplied by –1. To separate the
single-valued continuous branches of the function it suffices to
make two cuts respectively from the point and from the point
to infinity (the proof is the same as for the function The scheme
of the function is shown in Figure 74 (the Riemann surface is
shown in Figure 120a).
b) See Figure 75 (the Riemann surface is shown in Figure 120b). Hint.
FIGURE 74
FIGURE 75
304. See 303. a) Since during a turn around
each of the points and the value of the function
is multiplied by The scheme sought is shown
in Figure 76 (the Riemann surface is shown in Figure 121).
b) After a turn around the point the value of the function
turns out to be multiplied by During a turn around the
point varies by and varies by
i.e., the value of the function is multiplied by The scheme