1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 31

2. If z = J3-i, we haver= .J3+T = 2, cosB = J3/2, sinB = -1/2,
   so that 00 = -7r /6. Hence

J3 -i = 2 [cos ( ~7r ) + i sin ( - i)]

= 2[cos(7r/6)-isin(7r/6)]

The polar form of the conjugate of z = r( cos B + i sin B) is given by

z =a - bi= r( cos B - i sin B) (1.8-3)

If the polar coordinates are taken about a point z 0 f. O, wit.h the polar

line zoX' parallel to OX, with the same orientation (as in Fig. 1.3), the

polar representation of a point z f. z 0 is given by

z - z 0 = r( cos B + i sin B)

where r = lz - z 0 1, and B is the measure of the angle from the polar line
to the vector joining z 0 to z.

Theorem 1.6 Two nonzero complex numbers in polar form are equal iff
their moduli are equal and their arguments differ by a multiple of 27r.

Proof Let

If r 1 = r 2 and 01 = 02 + 2k7r (k an integer), it follows at once that
Rez1 = Rez2 and Imz1 = Imz2, so that z1 = z2.
Conversely, suppose that z1 = z 2. Then we have lz1I = lz 2 1, or r1 =
r 2 , and

or

y

zo~--'--------<x, L

(^0) x
Fig. 1.3