Complex Analysis - Maths KU

Answers to Selected Odd-Numbered Problems ANS-3
13. 2+2 √ 3 i 15. − 5√3 5
−
2 2 i
√
2
17. 5.5433 + 2.2961i 19. 8i;
4 −
√
2
4 i
21. 30 √ �
2 cos π π
�
+ i sin ≈ 40.9808 + 10.9808i
12 12
√ �
2
23. cos
2
5π
�
5π
+ i sin = −
4 4
1 1
−
2 2 i
1
25. −512 27. 32 i
�
29. −64i 31. 32 cos 13π
�
13π
+ i sin =16
6 6
√ 3+16i
33. cos 2θ = cos 2 θ − sin 2 θ, sin 2θ = 2 sin θ cos θ
35. n =6
Exercises 1.4, page 26
In answers 1–13, the principal nth root is given first.
1. w0 =2, w1 = −1+ √ 3i w2 = −1 − √ 3i
3. w0 =3i, w1 = −3i
5. w0 =
√ 2
2 +
√
2
i, w1 = −
2
√
2
2 −
√
2
2 i
7. w0 = 1
3√ +
2 1
3√ i, w1 ≈−1.0842 + 0.2905i, w2 ≈ 0.2905 − 1.0842i
2
√
2
9. w0 =
2 +
√ √
6
2
i, w1 = −
2 2 −
√
6
2 i
11. w0 =2+i, w1 = −2 − i
13. w0 ≈ 1.3477 + 0.1327i, w1 ≈ 0.8591 + 1.0469i,
w2 ≈−0.1327 + 1.3477i, w3 ≈−1.0469 + 0.8591i,
w4 ≈−1.3477 − 0.1327i, w5 ≈−0.8591 − 1.0469i,
w6 ≈ 0.1327 − 1.3477i, w7 ≈ 1.0469 − 0.8591i
15. (b) 4+3i, −4 − 3i
√ √
2
2
17. ± (1 + i); ± (1 − i)
2 2
19. (b) n =3: 1, − 1
2 +
√
3 1
i, −
2 2 −
√
3
2 i;
n =4:1,i,−1, −i;
n =5:
1, 0.3090+0.9511i, −0.8090+0.5878i, −0.8090−0.5878i, 0.3090−0.9511i,
√
2
25. (c) −
2 +
√
2
2 i,
√
2
2 −
√
2
2 i