Complex Analysis - Maths KU

Answers to Selected Odd-Numbered Problems ANS-17
Exercises 5.5, page 281
1. 8πi 3. −2πi
5. −π(20 + 8i) 7. (a) −2π (b) 2π
9. −8π 11. −2πe −1 i
13. 4
3 πi
15. (a) −5πi (b) −5πi (c) 9πi (d) 0
17. (a) −π(3 + i) (b) π(3 + i) 19. π � 8
3 +12i�
21. 0 23. −πi
25. 6
27. (a) 16; 4 (b) 25; 9 (c) 7;3
Exercises 5.6, page 294
5. f(z) = cos θ0+i sin θ0 = e iθ0 , g(z) =f(z) = cos θ0−i sin θ0 = e −iθ0 is constant
and so is analytic everywhere.
7. f(z) =2¯z +3i, g(z) =f(z) =2z − 3i is a polynomial function and so is
analytic for all z.
9. F(x, y) =(x 2 − y 2 − 2xy)i +(y 2 − x 2 − 2xy)j
11. F(x, y) =(e x cos y)i − (e x sin y)j
13. Ω(z) = e −iθ0 z; equipotential lines are the family of straight lines
x cos θ0 + y sin θ0 = c1; the streamlines are the family of straight lines
−x sin θ0 + y cos θ0 = c2.
15. Ω(z) =z 2 −3iz; equipotential lines are the family of hyperbolas x 2 −y 2 +3y =
c1; the streamlines are the family of hyperbolas 2xy − 3x = c2.
17. F(x, y) =−2xyi +(y 2 − x 2 )j
21. (a) For a point (x, y) far fromthe origin, the velocity field is given by
F(x, y) ≈ Ai, that is, the flow is a nearly uniform.
23. (a) The streamlines are Arg(z − x1) = c1, which are rays with vertex at
z = x1.
25. Circulation is 0; net flux is 0.
27. Circulation is 0; net flux is 2π.
29. Circulation is −4π; net flux is 12π.
Chapter 5 Review Quiz, page 297
1. false 3. true
5. true 7. true
9. true 11. true
13. false 15. true
17. true 19. true