Complex Analysis - Maths KU

7.2 Linear Fractional Transformations 401
Moreover, from (24) of Section 2.6 we have that
lim
z→∞
az + b
cz + d
a/z + b a + zb
= lim = lim
z→0c/z
+ d z→0c
+ zd
= a
c .
The values of these two limits indicate how to extend the definition of T .In
particular, if c �= 0, then we regard T as a one-to-one mapping of the extended
complex plane defined by:
⎧
az + b
, z �= −d ,z �= ∞
⎪⎨
cz + d c
T (z) = ∞, z = −
⎪⎩
d
c
a
, z = ∞ .
c
A special case of (3) corresponding to a =0,b =1,c = 1, and d =0
is the reciprocal function defined on the extended complex plane. Refer to
Definition 2.7.
EXAMPLE 1 A Linear Fractional Transformation
Find the images of the points 0, 1 + i, i, and ∞ under the linear fractional
transformation T (z) =(2z +1)/ (z − i).
Solution For z = 0 and z =1+i we have:
T (0) =
2(0) + 1
0 − i
1
+ i)+1 3+2i
= = i and T (1 + i) =2(1 =
−i (1 + i) − i 1 =3+2i.
Identifying a =2,b =1,c = 1, and d = −i in (3), we also have:
�
T (i) =T − d
�
= ∞
c
and T (∞) = a
c =2.
Circle-Preserving Property In the discussion preceding Example
1 we indicated that the reciprocal function 1/z is a special case of a linear
fractional transformation. We saw two interesting properties of the reciprocal
mapping in Section 2.7. First, the image of a circle centered at the pole
z = 0 of 1/z is a circle, and second, the image of a circle with center on the
x- ory-axis and containing the pole z = 0 is a vertical or horizontal line.
Linear fractional transformations have a similar mapping property. This is
the content of the following theorem.
(3)