Complex Analysis - Maths KU

Chapter 1 Review Quiz 47
27. The principal argument of z = −1 − i is .
28. ¯z 2 1 +¯z2 2
= .
29. arg � (1 + i) 5� = , � � (1 + i) 6�� = ,Im � (1 + i) 7� and Re
= ,
� (1 + i) 8� = .
30.
�483 = .
� 1
2 + √ 3
2 i
31. If z is a point in the second quadrant, then i¯z is in the quadrant.
32. i 127 − 5i 9 +2i −1 = .
33. Of the three points z1 =2.5 +1.9i, z2 =1.5 − 2.9i, and z3 = −2.4 +2.2i,
is the farthest from the origin.
34. If 3i¯z − 2z = 6, then z = .
35. If 2x − 3yi +9=−x +2yi +5i, then z = .
5
36. If z =
− √ , then Arg(z) = .
3+i
37. If z �= 0 is a real number, then z+z −1 is real. Other complex numbers z = x+iy
for which z + z −1 is real are defined by |z| = .
38. The position vector of length 10 passing through (1, −1) is the same as the
complex number z = .
39. The vector z =(2+2i)( √ 3+i) lies in the quadrant.
40. The boundary of the set S of complex numbers z satisfying both Im(y) > 0
and | z − 3i | > 1is .
41. In words, the region in the complex plane for which Re(z) < Im(z) is .
42. The region in the complex plane consisting of the two disks |z + i| ≤1 and
|z − i| ≤1is (connected/not connected).
43. Suppose that z0 is not a real number. The circles |z − z0| = |¯z0 − z0| and
|z − ¯z0| = |z0 − ¯z0| intersect on the (real axis/imaginary axis).
44. In complex notation, an equation of the circle with center −1 that passes
through 2 − i is .
45. A positive integer n for which (1 + i) n = 4096 is n = .
46.
�
�
� (4 − 5i)658
� (5 + 4i) 658
�
�
�
� = .
47. From (cos θ + i sin θ) 4 = cos 4θ + i sin 4θ we get the real trigonometric identities
cos 4θ = and sin 4θ = .
48. When z is a point within the open disk defined by |z| < 4, an upper bound for
�
� z 3 − 2z 2 +6z +2 � � is given by .
49. In Problem 20 in Exercises 1.6 we saw that if z1 is a root of a polynomial
equation with real coefficients, then its conjugate z2 =¯z1 is also a root. Assume
that the cubic polynomial equation az 3 + bz 2 + cz + d = 0, where a, b, c, and
d are real, has exactly three roots. One of the roots must be real because
.
50. (a) Interpret the circular mnemonic for positive integer powers of i given in