Complex Analysis - Maths KU

–3
y
3
2
1
–2 –1 1 2 3
–1
–2
–3
(a) Vertical lines in the z-plane
v
20
15
10
5
–20 –15 –10 –5
–5
–10
–15
w = z 2
5 10 15 20
–20
(b) The images of the lines in (a)
Figure 2.20 The mapping w = z 2
x
u
2.4 Special Power Functions 83
shown in color in Figure 2.19(a) is mapped onto the set consisting of the point
w = 0 together with the negative u-axis shown in black in Figure 2.19(b).
In order to gain a deeper understanding of the mapping w = z 2 we next
consider the images of vertical and horizontal lines in the complex plane.
EXAMPLE 2 Image of a Vertical Line under w = z2 Find the image of the vertical line x = k under the mapping w = z2 .
Solution In this example it is convenient to work with real and imaginary
parts of w = z 2 which, from (1) in Section 2.1, are u(x, y) =x 2 − y 2 and
v(x, y) =2xy, respectively. Since the vertical line x = k consists of the points
z = k + iy, −∞