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

Solutions 189
sum of multi-valued functions we have where is one of
the values of and is one of the values of Since for the
functions and the statement of the problem holds, there exist
two continuous images and starting respectively
from the points and If and are the parametric equations
of the curves and then the function (which
is continuous, being the sum of continuous functions) is the parametric
equation of the required curve because
In an identical way one considers the case in which
(in this last case the continuous
function sought is because by hypothesis the curve
C does not pass through the points at which the function is not
defined, and consequently
4) Suppose that and that for the statement of the
problem is true. By the definition of the function we have
where is one of the values of The mapping can be
considered as the composition of two mappings, and
Since for the function the statement of the problem holds, there
exists at least one continuous image of the curve C under the mapping
beginning at the point By virtue of the result of Problem
293, there exists at least one continuous image of the curve under
the mapping beginning at the point The curve is the
curve required.
313. At the point chosen arbitrarily, the function takes
values: where Since
the sum of continuous functions is a continuous function, the single-valued
continuous branches of the function are the following functions:
where
314. a) See Figure 90. Hint. Use the schemes of the Riemann surfaces
of the functions and (cf., 288, 299). b) See Figure 91. Hint.
Cf., 304, 307. c) See Figure 92. Hint. Cf., 288, 292. d) See Figure
93 (the Riemann surface is shown in Figure 127). Hint. Draw first the
schemes of the Riemann surfaces of the functions and