Complex Analysis - Maths KU

406 Chapter 7 Conformal Mappings
Recall, a circle is uniquely determined
by three noncolinear points.
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method to construct a linear fractional transformation w = T (z), which maps
three given distinct points z1, z2, and z3 on the boundaryof D to three given
distinct points w1, w2, and w3 on the boundaryof D ′ . This is accomplished
using the cross-ratio, which is defined as follows.
Definition 7.3 Cross-Ratio
The cross-ratio of the complex numbers z, z1, z2, and z3 is the complex
number
z − z1 z2 − z3
. (11)
z − z3 z2 − z1
When computing a cross-ratio, we must be careful with the order of the
complex numbers. For example, you should verify that the cross-ratio of 0, 1,
i, and 2 is 3 1
1 1
4 + 4i, whereas the cross-ratio of 0, i, 1,and2is4 − 4i. We extend the concept of the cross-ratio to include points in the extended
complex plane byusing the limit formula (24) from the Remarks in Section
2.6. For example, the cross-ratio of, say, ∞, z1, z2, and z3 is given bythe
limit
z − z1 z2 − z3
lim
.
z→∞z
− z3 z2 − z1
The following theorem illustrates the importance of cross-ratios in the studyof
linear fractional transformations. In particular, we prove that the cross-ratio
is invariant under a linear fractional transformation.
Theorem 7.4 Cross-Ratios andLinear
Fractional Transformations
If w = T (z) is a linear fractional transformation that maps the distinct
points z1, z2, and z3 onto the distinct points w1, w2, and w3, respectively,
then
z − z1 z2 − z3
=
z − z3 z2 − z1
w − w1 w2 − w3
(12)
w − w3 w2 − w1
for all z.
Proof Let R be the linear fractional transformation
z − z1 z2 − z3
R(z) = ,
z − z3 z2 − z1
(13)
and note that R(z1) =0,R(z2) = 1, and R(z3) =∞. Consider also the linear
fractional transformation
z − w1 w2 − w3
S(z) = . (14)
z − w3 w2 − w1