Complex Analysis - Maths KU

1.4 Powers and Roots 23
arg(z) =π/4 and arg(¯z) =−π/4. Student B disagrees because he feels that
he has a counterexample: If z = i, then ¯z = −i; we can take arg(i) =π/2
and arg(−i) =3π/2 and so arg(i) �= − arg(−i). Take sides and defend your
position.
46. Suppose z1, z2, and z1z2 are complex numbers in the first quadrant and that
the points z =0, z =1, z1, z2, and z1z2 are labeled O, A, B, C, and D,
respectively. Study the formula in (6) and then discuss how the triangles OAB
and OCD are related.
47. Suppose z1 = r1 (cos θ1 + i sin θ1) and z2 = r2 (cos θ2 + i sin θ2). If z1 = z2,
then how are r1 and r2 related? How are θ1 and θ2 related?
48. Suppose z1 is in the first quadrant. For each z2, discuss the quadrant in which
z1z2 could be located.
(a) z2 = 1
2 +
√
3
i
2
√
3 1
(b) z2 = − +
2 2 i
(c) z2 = −i (d) z2 = −1
49. (a) For z �= 1, verify the identity
1+z + z 2 + ···+ z n =
1 − zn+1
.
1 − z
(b) Use part (a) and appropriate results from this section to establish that
1 + cos θ + cos 2θ + ···+ cos nθ = 1
2 + sin � n + 1 �
θ 2
sin 1
2 θ
for 0