Review Exercises TEST 1 MA 437/537 Spring 2014
1. Take-home quiz 1, homework problems
Sketch the set of points z in the complex plane determined by the given condition. Is the set
open, closed, bounded, connected, a domain? What is the closure of the set?
3) |z + 2i| = |z − 2i|
2) Re(z + 2z) > 1
4) Im(z) < 3
5)
6) |Im(2z + 1)| < 1
8) 2Imz = |z|2
−π
6
< Argz ≤ π2
7) {2 < |z − i| ≤ 4} ∪ {Imz > 0}
9) |z − 2 + i| = 5
10) Re(z + 1 + 2i) ≥ 3
12) {z : |z| < 5 and
11) |z − i| > 3
2Re z ≤ Im z}
13. Verify that z = −i satisfies the equation z 3 + z = 0.
14. Solve for z
z 2 = −i
15. Solve for z
z4 = i
16. Find (8i)1/3
17. Find (−i)1/2
18. Find (1 − i)1/2
19. Find the following:
(1 − i)7
and
arg(1 − i)7
Find the argument arg(z) of each of the following complex numbers and write each in polar
form
i
20) 1 − i
21) 1 −
i
22) (1 − i)2007
23)
1−i
1+i
24. Find
Arg
25. Find
arg
1+i
1−i
2i
1+i
26. Find Arg z of the following complex number z and write z in polar form
z=
i7 (1 − i)
1+i
27. Show that
arg z1 z2 = arg z1 + arg z2
Give an example
Argz1 z2 6= Argz1 + Argz2
28. Show that
e2iθ = e−2iθ
29. Use de Moivre’s formula to derive the following identity:
sin 3θ = 3 sin θ − 4 sin3 θ
30. Use the formula eiθ = cos θ + i sin θ to derive the following equality:
e−iθ = cos θ − i sin θ
Find |e−iθ |.
31. Show that
|3z − 5z 3 + 1| ≤ 9
if |z| ≤ 1.
32. Show that
|e−2z−100i | ≤ e6
if |z| ≤ 3.
33. Prove that
|z1 + z2 | ≤ |z1 | + |z2 |
34. Prove that
|z1 z2 | = |z1 ||z2 |
35. Show that
| − z 2 − z − 1| ≤ 7,
if |z| ≤ 2.
36. Prove that
|Rez| + |Imz| ≤
√
2|z|
37. Give definitions a) G is a domain b) f (z) is continuous at a point z0
38. Find the domain of definition of the following function
f (z) =
3z + 1
z−z
39. Show that
Re(iz) = −Imz
Im(iz) = Rez
40. Locate the numbers z1 + z2 and 2z1 − 3z2 , where z1 = 2 − 3i and z2 = −5i.
41. Prove that
z1 z2 = z1
z2
42. Prove that z either real or pure imaginary iff
z2 = z2
43. Prove that nonzero complex numbers z1 and z2 have the same moduli iff there are complex
numberc1 and c2 such that z1 = c1 c2 and z2 = c1 c2 .
44. Show that the hyperbola x2 − y 2 = 1 can be written
z2 + z2 = 2