1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

28 CHAPTER 1 • COMPLEX NUMBERS

from the first and second sets, respectively. In this case, arg z 1 + arg z2 =
{ B1 + B2 : B1 E arg z1 and B2 E arg z2}.

Using Equality (1-35) gives z-^1 = ~ = ~ = ~e-i^8. In other words,

z -1 = -1 [ cos ( -u ") +ism.. ( -u ")] = -e^1 -i9.

r r

Recalling that cos (-B) = cos (B) and sin (-B) =- sin (B), we also have

z = r (cosB -isinB) = r [cos (-B) + isin (-B)] = re-i^9 , and

Z t = ri [cos(B1-B2)+ isin(B1 -82)] = rie;(e,-e,>_

z2 r2 r2

If z is in the first quadrant, the positions of t he numbers z, z, and z-^1 are
as shown in Figure 1.14 when [z[ < 1. Figure 1.15 depicts the situation when
[z[ > 1.

y

i=(O, I)

The unit circle

Figure 1.14 Relative positions of z, z,

and z-^1 when jz[ < 1.

y

(

The unit circle

Figure 1.15 Relative positions of z, z,

and z-l when [z[ > 1.