5. Let Ω ⊆ C be open and a ∈ Ω. Suppose that f is analytic on Ω except for an isolated singularity at a. Extend Theorem 5.2.6 by showing that f has a pole of order m ∈ N at a if and only if there is a...

5. Let Ω
⊆
C be open and
a

∈
Ω. Suppose that
f
is analytic on Ω except for an isolated singularity at
a. Extend Theorem 5.2.6 by showing that
f
has a pole of order
m

∈
N at
a
if and only if there is a function
g

∈

H(Ω) such that
g(a)
/= 0 and

g(z)

f
(z) =

(z

−

a)m

for all
z

∈
Ω
\ {
a
}.