Math 411: Honours Complex Variables - University of Alberta

64 CHAPTER 9. HOLOMORPHIC FUNCTIONS ON ANNULI
(a) On A0,1(0): For |z|< 1, we have
and, for |z|< 3,
1
3−z =
We thus have for z ∈ A0,1(0) that
f(z) =
(b) On A1,3(0): For |z|> 1, we have
1
1−z
so that, for z ∈ A1,3(0):
1
1−z =
n�
z n
n=0
1
3 � 1− z
� =
3
1
3
∞�
n=0
∞�
n=0
�
z
�n .
3
�
1− 1
3n+1 �
z n .
1 1
= − = −
z −1 z � 1− 1
∞� 1
� = −
z
z
n+1,
f(z) = −
(c) On A3,∞(0): For |z|> 3, we have
− 1
3−z
and thus, for z ∈ A3,∞(0):
� ∞�
n=1
= 1
z −3 =
f(z) =
1
+
zn ∞�
n=0
1
z � 1− 3
� =
z
n=0
zn 3n+1 �
.
∞�
(3 n−1 −1) 1
zn. n=1
∞�
n=0
3 n
z n+1