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4.5. The Fundamental Theorem: Functions of a Complex Variable 143

to consider how the image point h(z) moves correspondingly through the
w-plane. Thus, as z traces out a certain curve, one would expect h(z) to
trace out a curve in the w-plane. By looking at the images of special curves,
we can analyze the behaviour of h. In particular, if a curve in the z-plane
passes through a zero of h(z), its image in the w-plane must pass through
the origin. Thus, the problem of showing that h has a zero reduces to the
problem of showing that some image curve must pass through the origin.

Exercises

1. Let h(z) = .z~ - (3 + i)% + (1 - 29.

(a) Draw the real and imaginary axes of two complex planes, to be

labelled the t-plane and the w-plane. In the r-plane plot each

of the points 0, 1, i, 1 + i, 5 + i, -2, and in the w-plane plot the

images of these points under h.

(b) Let x be real, and show that h(x) = (x2 - 3x + 1) - (x + 2)i.

Show that the image of the real axis under the mapping h is the

curve given parametrically in the w-plane by

u=x2-3x+1

v = -(x + 2).

Verify that this image curve is a parabola with equation u =

v2 + 7v + 11. Sketch this parabola, and indicate by an arrow the

direction in which h(z) moves along the parabola as z moves in

the positive direction along the real axis.

2. Consider the polynomial p(z) = 2’ + z + 1. This polynomial has two
   zeros, both on the circle in the r-plane of radius 1. We study the
   impact of this fact on the images of circles of various radii under the
   mapping z - p(z).

(a) Verify that

p(rcos0 + irsine) = r(cos0 + isinO)(2rcose+ 1) + (1 - r”).

Deduce that, for any point Z, its image p(z) in the zu-plane can be

found by following a dilatation with center 0 and magnification

factor (2r cosfl + 1) by a translation 1 - r2 in the direction of

the positive real axis.

(b) Verify that p(0) = 1 and that

~p(rcose+irsine) - 11 < r(2r+ l)+r2 = r(3r+ 1).