Complex Analysis - Maths KU

Answers to Selected Odd-Numbered Problems ANS-7
Chapter 2
Exercises 2.1, page 56
1. (a) 6i (b) 2 (c) 39− 28i
3. (a) 0 (b) loge 4+ 1
1
πi (c) 2 2 loge 2+ 1
4 πi
5. (a) 3i (b) −12 + 13i (c) −24 + 4i
7. (a) 3+i (b) 2 (c) √ 5+ 4
5 i
9. u =6x − 5; v =6y +9
11. u = x 3 − 2x − 3xy 2 +6;v =3x 2 y − 2y − y 3
13. u = x2 + x − y 2
(x +1) 2 2xy + y
; v = −
+ y2 (x +1) 2 + y2 15. u = e 2x cos(2y + 1); v = e 2x sin(2y +1)
17. u = r cos θ; v = −r sin θ
19. u = r 4 cos 4θ; v = r 4 sin 4θ
21. u = e r cos θ cos (r sin θ); v = e r cos θ sin (r sin θ)
23. C 25. all z such that z �= 1
Exercises 2.2, page 65
1. the horizontal line v = −3 3. the half-plane Im(w) > 6
5. the line v =4− u 7. the half-plane Re(w) ≥ 3
9. the parabola u = 1
4 v2 − 1 11. the ray −∞