Complex Analysis - Maths KU

7.2 Linear Fractional Transformations 405
A −1 ⎛ ⎞
a
= ⎝
b
⎠
c d
Byidentifying e =
matrix
−1
=
⎛
1
⎝
ad − bc
d −b
⎞
⎠ .
−c a
1
ad − bc in (8) we can also represent T −1 (z) bythe
⎛ ⎞
⎝
d −b
⎠ .
−c a
∗ (10)
EXAMPLE 4 Using Matrices
Suppose S(z) =(z− i)/ (iz − 1) and T (z) =(2z−1)/ (z + 2). Use matrices
to find S−1 (T (z)).
Solution We represent the linear fractional transformations S and T bythe
matrices
⎛ ⎞ ⎛ ⎞
1
⎝
−i
⎠ and
2
⎝
−1
⎠,
i −1
1 2
respectively. By (10), the transformation S−1 is given by
⎛ ⎞
−1
⎝
i
⎠,
−i 1
and so, from (9), the composition S−1 ◦ T is given by
⎛ ⎞ ⎛ ⎞ ⎛
⎞
−1
⎝
i 2
⎠ ⎝
−1 −2+i
⎠ = ⎝
1+2i
⎠
−i 1 1 2 1 − 2i 2+i
Therefore,
S −1 (T (z)) =
(−2+i) z +1+2i
(1 − 2i) z +2+i .
Cross-Ratio In applications we often need to find a conformal mapping
from a domain D that is bounded bycircles onto a domain D ′ that is bounded
bylines. Linear fractional transformations are particularlywell-suited for such
applications. However, in order to use them, we must determine a general
∗ You may recall that this matrix is called the adjoint matrix of A.