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

182 Problems of Chapter 2
232). Since the value of at the initial point of the curve can be
chosen in different ways, we have exactly continuous curves, images
of the curve under the mapping (the uniqueness may be
lost only at the point but the curve does not pas through this
point).
Answer. The continuous images are the curves
297. Let be an arbitrary curve joining the point 1 to an arbitrary
point without crossing the cut. From the solution of Problem 296, we
obtain that if the value of the function at the initial point of this curve is
multiplied by then the values at the final point, defined by continuity,
turns out to be multiplied by Hence
Answer.
298. Solving Problem 297 we have found that the branches of
the function are related each other in this way:
The unique branch point of the function is the
point During a simple turn around this point the argument of the
function changes by (cf., 294), i.e., the value of the function
varies by Consequently the scheme of the Riemann surface of the
function has the form shown in Figure 71 (the Riemann surface for
is shown in Figure 119).
FIGURE 71
299. During a turn around the point varies by
and it does not change during any other turn around the other points
(along sufficiently small circles). During a turn around the point
thus changes by and it remains constant during any other