Math 411: Honours Complex Variables - University of Alberta

Define
⎧
⎪⎨
g: D ∪{z0} → C, z ↦→
⎪⎩
∞�
an+2(z −z0) n , z ∈ Bǫ(z0),
n=0
Then g is a holomorphic function extending f.
f(z), z ∈ D \Bǫ(z0).
Definition. Let D ⊂ C be open, let f: D → C be holomorphic, and let z0 ∈ C\D
be an isolated singularity of f. Then z0 is called a pole of f if lim|f(z)|=
∞.
z→z0
Example. For n ∈ N, the function
has a pole at 0.
C\{0} → C, z ↦→ 1
z n
Theorem 8.2 (Poles). Let D ⊂ C be open, let f : D → C be holomorphic, and let
z0 ∈ C\D be an isolated singularity of f. Then z0 is a pole of f ⇐⇒ there exist a
unique k ∈ N and a holomorphic function g: D ∪{z0} → C such that g(z0) �= 0 and
for z ∈ D.
f(z) = g(z)
(z −z0) k
Proof.
”⇐” This follows directly from the definition of a pole.
”⇒” Let us prove the uniqueness first. Suppose that there exist natural numbers
k1 ≤ k2 and holomorphic functions g1,g2: D ∪{z0} → C such that gj(z0) �= 0 and
f(z) = gj(z)
(z −z0) kj
for z ∈ D and j = 1,2. If k2 > k1, we then find for all z ∈ D that
g2(z0) = lim
z→z0
g2(z) = lim(z
−z0)
z→z0
k2−k1g1(z) = 0·g1(z0) = 0,
which is a contradiction. Hence k1 = k2 and thus g1 = g2 on D and, by continuity,
on D ∪{z0}.
To establish the existence of k and g, choose r > 0 such that Br(z0)\{z0} ⊂ D
and |f(z)|≥ 1 for all z ∈ Br(z0)\{z0}. Then
Br(z0)\{z0} → C, z ↦→ 1
f(z)
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