You may use the fact that if
f
0
(
z
) = 0
for all
z
∈
D
, then
f
is constant.
(b) Let
f
be an entire function and
M
be a positive constant (
M >
0) such that

f
(
z
)
 ≥
Me

y
for all
z
=
x
+
iy
∈
C
. Show that we can find a constant
k
∈
C
such
that
f
(
z
) =
ke
iz
for all
z
∈
C
.
(c) Establish the following integration formula with the aid of residues:
Z
∞
∞
x
sin
x
(
x
2
+ 1)(
x
2
+ 4)
dx
=
π
3
(
e

1

e

2
)
.
Complete explanations are required.
MATH2101
PLEASE TURN OVER
1
3. (a) State Cauchy’s integral formulas for
f
and its derivatives.
State Cauchy’s inequalities for
f
and its derivatives.
Make sure you explain all quantities and functions that appear.
(b) Let
f
be entire and assume that for some
M >
0 we have

f
(
z
)
 ≤
M
(1 +

z

)
1
/
3
,
∀
z
∈
C
.
Show that
f
is a constant function.
(c) Let
f
be holomorphic on an open set
U
containing the closed disc
D
(0
, R
). Let
C
be the circle centered at 0 with radius
R
traversed anticlockwise and let
z
∈
D
(0
, R
).
Show that
f
(
z
) +
f
(

z
)
2
=
1
2
πi
Z
C
f
(
w
)
w
w
2

z
2
dw,
f
0
(
z
)

f
0
(

z
)
2
=
1
2
πi
Z
C
f
(
w
)
2
wz
(
w
2

z
2
)
2
dw.
4. (a) Show that the map
w
=
T
(
z
) =
z

i
z
+
i
maps the upper halfplane
{
z
;
=
(
z
)
>
0
}
conformally onto the unit disc
{
w
;

w

<
1
}
.
(b) Explain why the map
w
=
T
(
z
) =
z

i
z
+
i
maps the first quadrant
{
z
;
<
(
z
)
>
0
,
=
(
z
)
>
0
}
conformally onto the lower halfdisc
{
w
;

w

<
1
,
=
(
w
)
<
0
}
.
(c) Find a conformal map from the first quadrant
{
z
;
<
(
z
)
>
0
,
=
(
z
)
>
0
}
onto the
upper halfplane
{
z
;
=
(
z
)
>
0
}
.
You've reached the end of your free preview.
Want to read all 3 pages?
 Fall '18