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

The complex numbers 79
C from to the function varies continuously from to
One can also use this property conversely, i.e., for the definition of the
function
Indeed, suppose at a certain point one of the values of the
function be chosen. Let C be a continuous curve beginning at
and ending at a certain point Moving along the curve C we choose for
every point lying on C one of the values of the function in such a
way that these values vary continuously while we move along the curve C
starting from the value So when we arrive at the point the value
is completely defined. We say that is the value of
defined by continuity along the curve C under the condition
If the values of the function chosen for all points of the curve C, are
represented on the plane then we obtain a continuous curve beginning
at the point and ending at the point This curve is one of the
continuous images of the curve C under the mapping
278. For the function let us choose
Define by continuity along the following curves: a) the
upper semi-circle of radius 1 with centre at the origin of the coordinates;
b) the lower semi-circle (Figure 28).
FIGURE 28 FIGURE 29
In fact, using for a function the definition by continuity along a certain
curve we may encounter some difficulties. Consider the following example.
279. Find all continuous images of a curve C with parametric
equation (Figure 29) under the mapping beginning:
a) at the point b) at the point