Complex Analysis - Maths KU

Answers to Selected Odd-Numbered Problems ANS-9
Exercises 2.3, page 76
1. |w − 3i| ≤1 3. |w| ≤3
5. |w + i| ≤2 7. triangle with vertices 2i, 1+2i, and 3i
√
2i
9. triangle with vertices 0, 1
2
√ 2+ 1
2
√ 2i, and − 1
2
11. triangle with vertices i, −3+i, and −2i
√ 2+ 1
2
13. f(z) =T ◦ M ◦ R(z) where R(z) =e πi/2 z, M(z) =3z, and T (z) =z +4
15. f(z) = T ◦ M ◦ R(z) where R(z) = e πi z, M(z) = 1
z, and
2
T (z) =z +1− √ 3i
17. f(z) =iz +2i 19. f(z) =e −πi/4 z + i
21. f(z) =z − 1,g(z) =iz
23. (a) w(t) =(z0 + b)(1 − t) +(z1 + b)t, 0 ≤ t ≤ 1, the line segment from
z0 + b to z1 + b
(b) w(t) =az0(1 − t)+az1t, 0 ≤ t ≤ 1, the line segment from az0 to az1
(c) w(t) =az0(1 − t)+az1t, 0 ≤ t ≤ 1, the line segment from az0 to az1
25. (a) f(z) =2e πi/4 z +1+i (b) f(z) =2e πi/4 z +1+i
(c) f(z) =2e πi/4 z +1+i
Exercises 2.4, page 97
1. arg(w) = 2
1
π 3. u =9− 3 36 v2 , −∞