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

Solutions 179
to the condition coincides, as we easily see, with the value of
defined by continuity along the curve according to the condition
Hence the value of for every outside the cut is equal to
287. Let be a continuous curve joining the points and and not
crossing the cut (Figure 68). Let be the value of the function
defined by continuity along the curve according to the condition
Since the curve does not pass through the cut the values
and correspond to the same branch of the function The curve
turns once around the point The values and are thus
different (cf., 283). Since and correspond to the same branch of
the function and correspond to different branches.
288. If then the circle with centre at the point and with
a sufficiently small radius does not turn at all around the point
Therefore the variation of vanishes, and consequently the
variation of is zero, i.e., the value of does not change. The
variation of along a circle with centre at the point is equal
to In this case the variation of is equal to The value of
during a turn around the point thus changes into the opposite
value.
289. The curve is the image of the curve under the mapping
Consequently if is the variation of the argument along the
curve then (cf., 262 (b)), from which
290. If then the circle with centre at the point and with
a sufficiently small radius does not turn at all around the point
Consequently the variation of along this circle vanishes. But thus
the variation of also vanishes (cf., 289), i.e., the value of function
does not vary. This means that none of the points is a
branch point.
The variation of along a circle with centre at the point
is equal to Thus the variation of is equal to The value
of the function after a simple turn around the point turns
out to be multiplied by i.e., the point
is a branch point of the function
Answer.
291. Let be a continuous curve, not crossing the cut and joining
the point to the given point. Let be the continuous image of