This preview shows pages 1–2. Sign up to view the full content.
Extra Problems Sheet 7
Stephen Taylor
May 15, 2005
1. Prove that if
z
1
, z
2
, z
3
are distinct points and
w
1
, w
2
, w
3
are distinct points,
then the M¨
obius transformation
T
satisfying
T
(
z
1
) =
w
1
, T
(
z
2
) =
w
2
, T
(
z
3
) =
w
3
is unique.
A M¨
obius transformation given is given by
T
(
z
) =
az
+
b
cz
+
d
. Assuming
a
is not
zero, we ﬁnd the general form can be reduced to
T
(
z
) =
a
1
z
+
b
1
c
1
z
+
d
1
=
a

1
1
a

1
1
·
a
1
z
+
b
1
c
1
z
+
d
1
=
z
+
b
1
a

1
1
c
1
a

1
1
z
+
a
1
d
1
≡
z
+
a
bz
+
c
Substituting the three transformed points into this equation, we obtain the
system three equations
w
i
=
T
(
z
i
) =
z
i
+
a
bz
i
+
c
where
i
=
{
1
,
2
,
3
}
So we have a system of three linear equations with three unknowns, all of which
must be independent since all points are distinct. Therefore, provided a solution
exists, it must be unique.
±
2. Find all M¨
obius transformations that map the unit disk onto the right
halfplane.
Let
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Three '09
 Cong
 Math

Click to edit the document details