Complex Analysis - Maths KU

2.5 Reciprocal Function 105
of the ideal point. This “fills in” the missing point in the circle � �w − 1
�
�
2 = 1
2 .
Therefore, the vertical line x = 1 is mapped onto the entire circle � �w − 1
�
�
2 = 1
2
bythe reciprocal mapping on the extended complex plane. This mapping can
be represented byFigure 2.43 with the “hole” at w = 0 filled in.
Because the ideal point ∞ is on everyvertical line in the extended complex
plane, we have � that �the
image of anyvertical line x = k with k �= 0 is the
�
entire circle �
1 �
�w − �
2k � =
� �
�
�
1 �
�
�2k
� under the reciprocal function on the extended
complex plane. See Problem 23 in Exercises 2.5. In a similar manner, we can
also show that horizontal lines are mapped to circles by w =1/z. We now
summarize these mapping properties of w =1/z.
Mapping Lines to Circles with w =1/z
The reciprocal function on the extended complex plane maps:
(i) the vertical line x = k with k �= 0onto the circle
� �
�
�
1 �
�w − �
2k � =
� �
�
�
1 �
�
�2k
� , and (5)
(ii) the horizontal line y = k with k �= 0onto the circle
�
�
�
1
�w +
2k i
�
�
�
� =
� �
�
�
1 �
�
�2k
� . (6)
These two mapping properties of the reciprocal function are illustrated in
Figure 2.44. The vertical lines x = k, k �= 0, shown in color in Figure 2.44(a)
are mapped by w =1/z onto the circles centered on the real axis shown in
black in Figure 2.44(b). The image of the line x = k, k �= 0, contains the point
(1/k, 0). Thus, we see that the vertical line x = 2 shown in Figure 2.44(a)
y v
3
3
2
1
w = 1/z
x u
–3 –2 –1 1 2 3
–3 –2 –1 1 2 3
–1
–2
–3
(a) Vertical and horizontal lines (b) Images of the lines in (a)
Figure 2.44 Images of vertical and horizontal lines under the reciprocal mapping
2
1
–1
–2
–3