4. Prove the uniqueness of the Laurent series in Theorem 5.1.4. 5. Let a ∈ C and 0 r R ∞. Suppose that f ∈ H(A(a; r, R)) and |f (z)|≤ M for n=−∞ all z ∈ A(a; r, R) and some M ≥ 0. If...

4. Prove the uniqueness of the Laurent series in Theorem 5.1.4.

5. Let
a


∈

C and 0

r R

∞. Suppose that
f


∈


H(A(a;
r
,


R)) and
|
f

(z)|
≤


M

for

c
n(z

−

a)n
is the Laurent series of
f

on
A(a;
r
,


R), show that
|
c
n
|
≤


M
/
R
n

for all
n


≥

0 and
|
c
n
|
≤


M
/
r
n

for all
n


≤

0.