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Solutions Homework 12
(1) (Quals 1995) Let
f
be an entire function on
C
and assume that

f
(
z
)
 ≤
A

z

k
+
B
for some constants
A,B
, integer
k
and all
z
∈
C
. Prove that
f
is a polynomial.
Solution:
Let
R >
0 and
γ
R
=
Re
it
,
0
≤
t
≤
2
π
. From Cauchy’s integral formula we
get for

z

< R
f
(
n
)
(0) =
n
!
2
πi
Z
γ
R
f
(
z
)
z
n
+1
dz,
and thus

f
(
n
)
(0)
 ≤
2
πR
n
!
2
π
AR
k
+
B
R
n
+1
→
0
,
as
R
→ ∞
for all
n > k
. Thus
f
(
n
)
(0) = 0 for all
n > k
. Applying this to the power
series expandsion of
f
around
z
= 0 we see that
f
(
z
) =
∑
k
n
=0
c
n
z
n
.
(2) (Quals 1995) Let
G
⊂
C
be a connected open set and let
h
f
n
i
be a sequence of
holomorphic functions on
G
, which converges uniformly on every compact subset of
G
to a function
f
. Prove that
f
is holomorphic on
G
.
Solution:
Let
a
∈
G
and
r >
0 such that
B
(
a
;
r
)
⊂
G
. Then
h
f
n
i
converges
uniformly to
f
on
B
(
a
;
r
) and thus
f
is continuous on
B
(
a
;
r
). Let Δ
⊂
B
(
a
;
r
).
Then by Cauchy’s Theorem for starlike open sets we know that
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 Fall '11
 Schep

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