Complex Analysis - Maths KU

60 Chapter 2 Complex Functions and Mappings
y
(a) The half-plane S
S′
v
2
w = iz
(b) The image, S′, of the half-plane S
Figure 2.2 The mapping w = iz
2i
S
x
u
planes. Clearly, such an illustration would give no insight into how points in
the z-plane are mapped onto points in the w-plane by f.
EXAMPLE 1 Image of a Half-Plane under w = iz
Find the image of the half-plane Re(z) ≥ 2 under the complex mapping w = iz
and represent the mapping graphically.
Solution Let S be the half-plane consisting of all complex points z with
Re(z) ≥ 2. We proceed as illustrated in Figure 2.1. Consider first the vertical
boundaryline x =2ofS shown in color in Figure 2.2(a). For anypoint z on
this line we have z =2+iy where −∞