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

204 Problems of Chapter 2
where is the derivative of polynomial with respect to
(cf., 276). We have
Since the equation has four
roots of order 1, only the values can
be multiple roots of the equation (of order 2). Putting these
values in the equation
one obtains that they can be roots of order two if takes the values,
–16, –38, 38, 16, respectively.
Answer. The roots of order two are the values: for
for for for
344. Let Set and consider the
single-valued mapping of the plane onto the complex plane defined
by In the plane let C be a circle of radius with centre
at the point (Figure 113) and the image of the circle C under
the mapping Decompose the polynomial
into monomials of first degree (cf., 269). We obtain
where all values
are roots of the equation By a counterclockwise turn
along the circle C, the argument of the factor does not change
if lies outside the disc D, bounded by C, and increases by if
lies inside this disc. Therefore, going counterclockwise along the circle C,
the argument of the function increases by where is the
number of roots (taking into account their multiplicities) of the equation
which lie inside D. Consequently the curve the image of
the circle C under the mapping turns around the point
times (Figure 114).
FIGURE 113 FIGURE 114