Complex Analysis - Maths KU

z
c(z)
Figure 2.41 Complex conjugation
z¯
2.5 Reciprocal Function 101
z 2
z 3
1
y
(a) Points z 1 , z 2 , and z 3 in the z-plane
Figure 2.40 Inversion in the unit circle
z 1
1
w = e g(z2 )
x u
iθ 1
r
1
g(z 3 )
1
v
g(z 1 )
(b) The images of z 1 , z 2 , and z 3 in the w-plane
�
�
unit circle), we have that |g(z)| = �
1
� r eiθ
�
�
�
1
� = < 1. So, the image under g of
r
a point z outside the unit circle is a point inside the unit circle. Conversely,
if r 1, and we
r
conclude that if z is inside the unit circle, then its image under g is outside the
unit circle. The mapping w = eiθ /r is represented in Figure 2.40. The circle
|z| = 1 shown in color in Figure 2.40(a) is mapped onto the circle |w| =1
shown in black in Figure 2.40(b). In addition, w = eiθ /r maps the region
shown in light color in Figure 2.40(a) into the region shown in light grayin
Figure 2.40(b), and it maps the region shown in dark color in Figure 2.40(a)
into the region shown in dark grayin Figure 2.40(b).
We end our discussion of inversion in the unit circle byobserving from
(2) that the arguments of z and g(z) are equal. It follows that if z1 �= 0is
a point with modulus r in the z-plane, then g(z1) is the unique point in the
w-plane with modulus 1/r lying on a ray emanating from the origin making
an angle of arg(z0) with the positive u-axis. See Figure 2.40. In addition,
since the moduli of z and g(z) are inverselyproportional, the farther a point
z is from 0 in the z-plane, the closer its image g(z) isto0inthew-plane, and,
conversely, the closer z is to 0, the farther g(z) is from 0.
Complex Conjugation The second complex mapping that is helpful
for describing the reciprocal mapping is a reflection across the real axis. Under
this mapping the image of the point (x, y) is(x, −y). It is easyto verify
that this complex mapping is given bythe function c(z) =¯z, which we call the
complex conjugation function. In Figure 2.41, we illustrate the relationship
between z and its image c(z) in a single copyof the complex plane. By
replacing the symbol z with re iθ we can also express the complex conjugation
function as c(z) =re iθ =¯re iθ . Because r is real, we have ¯r = r. Furthermore,
from Problem 34 in Exercises 2.1, we have e iθ = e −iθ . Therefore, the complex
conjugation function can be written as c(z) =¯z = re −iθ .