262 Chapter^5
so there exists R 1 > 0 such that for lzl > Ri we have
Now, given M > 0, the inequality
IP(z)I > %1anlJzJn > Mholds whenever JzJ > (2M/lani)^1 fn = R2. Therefore, we have IP(z)J > M
for lzl > R = max(Ri, R2), which means that limz.....,. 00 P(z) = oo.
Because of this result we define P(oo) = oo, thus making w = P(z) a
continuous function in C*. Since IP(z)J > M > 0 for lzl > R, it follows
that a polynomial function has no zeros outside a certain circle of radius R.
Let A be an arbitrary but finite complex number, and suppose, as before,
that P(z) is of degree n ;::=: 1. Since P(z) - A is again a polynomial of
degree n, the equation
P(z) =A
has n solutions (not necessarily distinct) A1, A2, ... , An. These values are
called the A-points of the polynomial, in particular, the zero points (or
simply the zeros) if A = 0. On the other hand, the equation P(z) = oo
has just the solution z =·oo, since P(z) is finite for every finite value of z.
However, the solution z = oo is regarded as having multiplicity n.
Thus we see that every polynomial function w = P( z) of degree n ;::=: 1
maps the extended complex plane onto itself, and that every point w has
at most n distinct inverse images. If n > 1, there are some exceptional
points in the extended w-plane with less than n distinct inverse images,
one of those exceptional points being oo.
Example For w f. 0, oo, the function w = z^2 has two distinct inverse
images (the square roots of w). However, for either w = 0 or w = oo there
is just one distinct inverse image.
5.15 THE FUNCTION w = (z - a)n, n > 1
This function is equivalent to a special polynomial function of degree n,
as seen by expanding the right-hand side. We propose to investigate the
mapping defined by this particular function. According to what was stated
at the end of Section 5.14 for polynomials in general, the function w =
(z - a)n with n > 1, maps the extended z-plane onto the extended w-
plane, every point of the w-plane having n distinct inverse images,. in this
case excepting the two points w = 0 and w = oo, each of which has just one