Complex Analysis - Maths KU

Answers to Selected Odd-Numbered Problems ANS-13
23. 1 2 i
25. 8i
27. f is not analytic at z = − 1 1
+ 3 3 i
29. f is analytic for all z
Exercises 3.2, page 157
17. a =1,b =3 23. f ′ (z) =−e −x cos y + ie −x sin y
Exercises 3.3, page 162
9. f(z) =x + i(y + C); f(z) =x 2 − y 2 + i(2xy + C);
f(z) = log e (x 2 + y 2 )+i
�
tan −1 y
x
+ C
�
;
f(z) =e x (x cos y − y sin y)+ie x (x sin y + y cos y + C)
11. f(z) =xy + x +2y − 5+i � 1
2 y2 − 1
2 x2 + y − 2x +1 �
13. (b) f(z) =
y
x2 + i
+ y2 x
x 2 + y 2
Exercises 3.4, page 170
1. x = c1, y = c2
(c) f(z) = i
z
3. c1x = x 2 + y 2 , −c2y = x 2 + y 2 ; the level curves u(x, y) = 0 and
v(x, y) = 0 correspond to x = 0 and y = 0, respectively.
9. (a) φ(x) =−50x +50 (b) Ω(z) =−50x +50− 50yi
11. (a) φ(θ) = 120
π
120 120
θ (b) Ω(z) = θ −
π π loge r
Chapter 3 Review Quiz, page 172
1. false 3. true
5. true 7. true
9. true 11. true
13. −
2z +5i
15. 2+i
(z 2 +5iz − 4) 2
17. f ′ (z) = (y − 1)2 − (x − 1) 2
[(x − 1) 2 +(y − 1) 2 ]
19. constant
2 + i
21. v(x, y) =e −x (x cos y + y sin y)
2(x − 1)(y − 1)
[(x − 1) 2 +(y − 1) 2 ] 2