MAT 572 Homework 3 Due Monday,9/18
Always proveyour answers.
1.
Let
G
be a region, let
f
:
G
!
C
be dierentiable, and suppose that the range of
f
is
contained in a circle. Prove that
f
is constant.
2.
Let
G
=
C
n
(
1
;
0] and let
n
2
N
. Find all continuous functions
f
:
G
!
C
such
that
f
(
z
)
n
=
z
for all
z
2
G
.
3.
(i) Find a region
G
with the following properties:
(a) There is a continuous function
f
:
G
!
C
with
z
2
+
f
(
z
)
2
=1for
z
2
G
.
(b)
G
is maximal among regions satisfying (a).
(ii) For the region you found in part (i), nd all functions satisfying condition (a).
4.
Let
T
be a Mobius transformation. Recall that
H
=
z
2
C
Im (
z
)
>
0
and
D
=
z
2
C
j
z
j
<
1
. Prove the following.
(i)
T
(
R
)=
R
i there is
A
2
M
2
(
R
)with
T
=
T
A
.
(ii)
T
(
R
)=
R
and
T
(
H
)=
H
i for every
A
2
M
2
(
R
)with
T
=
T
A
,det(
A
)
>
0.
(iii) Find all Mobius transformations
T
with
T
(
D
)=
D
. (Hint: Find a Mobius transfor-
mation
S
with
S
(
H
)=
D
. Show that
STS
1
T
as in part (b)
g
gives all solutions.)
(iv) Find all Mobius transformations
T
with
T
(
D
)=
D
and
T
(0) = 0.
(v) Let
a
2
D
. Find all Mobius transformations
T
with
T
(
D
)=
D
and
T
(
a
)=
a
. (These
are the \rotations of
D
about
a
".) (Hint: Find a Mobius transformation
U
with
U
(
D
)=
D
and
U
(0) =
a
.)