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

170 Problems of Chapter 2
Answer. times.
259. Answer. a) 1; b) 0; c) 1; d) 2.
260. Solution. If and are two functions which describe the
continuous variation of the argument along the curves and then
as a function describing the continuous variation of the argument
along C on can take: in the case (a) in the case (b)
(cf., 226, 239). It follows that
c)
Answer. a) b)
261. Hint. (use de Moivre’s formula; cf., 227):
a) (a semi-circle of radius
b) (a circle of radius
c) (a circle of radius twice covered).
262. Hint. Use the result of Problem 260(a). Answer. a) b)
263. Let be the parametric equation of the curve C and
By hypothesis the variation of is equal to The
parametric equation of the curve is
The variation of the argument of is thus equal to (cf., 262(c)).
Hence the curve turns around the point times. Answer.
times.
264. By hypothesis the variation of the argument of is equal to
of the argument of to of the argument of to
and of the argument of to By the result of Problem
260 (a) one obtains:
a) The variation of the argument of
is equal to Answer. The curve turns
around the point times.
b) Answer. times.
c) Answer. times.
d) Answer.
times.