Complex Analysis - Maths KU

ANS-22 Answers to Selected Odd-Numbered Problems
11. w = cos πz
by entry H-4 of Appendix III.
2
� �1/2 1+z
13. w =
by entries H-5 and E-4 of Appendix III.
1 − z
� �1/2
π/z −π/z
e + e
15. w =
by entries H-6 and E-4 of Appendix III.
e π/z − e −π/z
Exercises 7.2, page 408
1. T (0) = ∞, T(1) = i, T (i) =1, T(∞) =0
3. T (0) = −1, T(1) = i, T (i) =∞, T(∞) =1
5. |w| ≥1 and u ≥ 1
7. u ≤ 0 and |w − 1| ≥2
2
9. v ≥ 0 and � �w − 1
�
� ≥ 2
1
11. v ≥ 0 and u ≤ 1
2
13. The image consists of a set of all points w = u + iv such that � �w + 1
�
� ≥ 3
2
3 and
v ≤ 0.
15. The image consists of a set of all points w = u + iv such that � �w + 1
�
� ≥ 20
9
20
and u ≥− 1
2 .
17. (a) S −1 (z) =
19. (a) S −1 (z) =
21. T (z) =
25. T (z) =
2z +2
−z +2
z +1
z − i
z − 2
z − 1
3z − 3i
(1 + 4i)z − (4 + i)
Exercises 7.3, page 418
1. the first quadrant u ≥ 0, v ≥ 0
(b) S −1 (T (z)) =
(b) S −1 (T (z)) = 3
z
23. T (z) = 2z
z + i
(1 + i)z − 1
2z + i
3. the region bounded by the ray u =0, 0 ≤ v