Complex Analysis - Maths KU

2.4 Special Power Functions 99
43. Find two sets in the complex plane that are mapped onto the ray arg(w) =π/2
by the function w = z 2 .
44. Find two sets in the complex plane that are mapped onto the set bounded by
the curves u = −v, u =1− 1
4 v2 , and the real axis v = 0 by the function w = z 2 .
45. In Example 2 it was shown that the image of a vertical line x = k, k �= 0, under
w = z 2 is the parabola u = k 2 − v2
. Use this result, your knowledge of linear
4k2 mappings, and the fact that w = − (iz) 2 �
to prove that the image of a horizontal
line y = k, k �= 0, is the parabola u = − k 2 − v2
4k2 �
.
46. Find three sets in the complex plane that map onto the set arg(w) =π under
the mapping w = z 3 .
47. Find four sets in the complex plane that map onto the circle |w| = 4 under the
mapping w = z 4 .
48. Do lines that pass through the origin map onto lines under w = z n , n ≥ 2?
Explain.
49. Do parabolas with vertices on the x-axis map onto lines under w = z 1/2 ?
Explain.
50. (a) Proceed as in Example 6 to show that the complex linear function
f(z) =az + b, a �= 0, is one-to-one on the entire complex plane.
(b) Find a formula for the inverse function of the function in (a).
51. (a) Proceed as in Example 6 to show that the complex function f(z) = a
+ b,
z
a �= 0, is one-to-one on the set |z| > 0.
(b) Find a formula for the inverse function of the function in (a).
52. Find the image of the half-plane Im(z) ≥ 0 under each of the following principal
nth root functions.
(a) f(z) =z 1/2
(b) f(z) =z 1/3
(c) f(z) =z 1/4
53. Find the image of the region |z| ≤8, π/2 ≤ arg (z) ≤ 3π/4, under each of the
following principal nth root functions.
(a) f(z) =z 1/2
(b) f(z) =z 1/3
(c) f(z) =z 1/4
54. Find a function that maps the entire complex plane onto the set
2π/3 < arg(w) ≤ 4π/3.
55. Read part (ii) of the Remarks, and then describe how to construct a Riemann
surface for the function f(z) =z 3 .
56. Consider the complex function f(z) = 2iz 2 − i defined on the quarter disk
|z| ≤2, 0 ≤ arg(z) ≤ π/2.
(a) Use mappings to determine upper and lower bounds on the modulus
of f(z) = 2iz 2 − i. That is, find real values L and M such that
L ≤ � � 2iz 2 − i � � ≤ M.
(b) Find values of z that achieve your bounds in (a). In other words, find z0
and z1 such that |f(z0)| = L and |f(z1)| = M.
57. Consider the complex function f(z) = 1
3 z2 +1−i defined on the set 2 ≤|z| ≤3,
0 ≤ arg(z) ≤ π.