Complex Analysis - Maths KU

106 Chapter 2 Complex Functions and Mappings
2
1
–2
(a) The semi-infinite strip S
v
2
1
w = 1/z
–1 1 2 3
S′
–1
–2
y
–1 1 2 3
–1
(b) The image of S
Figure 2.45 The reciprocal mapping
S
u
x
maps onto the circle centered on the real axis containing � 1
2 , 0� shown in
Figure 2.44(b), and so on. Similarly, the horizontal lines y = k, k �= 0, shown
in color in Figure 2.44(a) are mapped by w =1/z onto the circles centered
on the imaginaryaxis shown in black in Figure 2.44(b). Since the image of
the line y = k, k �= 0, contains the point (0, −1/k), we have that the line
y = 2 shown in Figure 2.44(a) is the circle centered on the imaginaryaxis
containing the point � 0, − 1
�
2 shown in Figure 2.44(b), and so on.
EXAMPLE 4 Mapping of a Semi-infinite Strip
Find the image of the semi-infinite horizontal strip defined by1 ≤ y ≤ 2,
x ≥ 0, under w =1/z.
Solution Let S denote the semi-infinite horizontal strip defined by1 ≤ y ≤ 2,
x ≥ 0. The boundaryof S consists of the line segment x =0,1≤ y ≤ 2,
and the two half-lines y = 1 and y =2,0≤ x