40 CHAPTER 1 • COMPLEX NUMBERS
to 7r, it is on leaf 2; between 7r and^3 ;, it is on leaf 3; and finally, fort between
(^3) ; and 2n, it is on leaf 4.
Note further that, at (0, 0) , the curve has crossed over itself (at points other
than those corresponding with t = 0 and t = 2n); we want to be able to distin-
guish when a curve does not cross over itself in this way. The curve C is called
s im ple if it does not cross over itself, except possibly at its initial and terminal
points. In other words, the curve C: z (t), for a :::; t :::; b, is simple provided that
z (t 1 )-# z (t2) whenever t1 -# t2, except possibly when ti =a and t2 = b.
- EXAMPLE 1.2 3 Show that the circle C with center zo = xo +iyo and radius
R can be parametrized to form a simple closed curve.
Solution Note that C: z (t) = (x 0 + R cost) + i (Yo+ Rsin t) = zo +&it, for
0 :::; t :::; 27r, gives the required parametrization.Figure 1.23 shows that, as t varies from 0 to 27r, the circle is traversed
counterclockwise. If you were traveling around the circle in this manner, its
interior would be on your left. When a simple closed curve is paramet rized in
this fashion, we say that the curve has a positive orientation. We will have
more to say about this idea shortly.
We need to develop some vocabulary that will help describe sets of points in
the plane. One fundamental idea is that of an c neighborhood of the point zo.
It is the open disk of radius c > 0 about zo shown in Figure 1.24. Formally, it
is the set of all points satisfying the inequality { z : lz - zol < c} and is denoted
D,, (.zo). That is,De (Zo) = {z : lz -zol < c}. (1-49)
yz (1t) z (0) = z (21t)