Math 411: Honours Complex Variables - University of Alberta

Proposition 10.2 (Winding Numbers AreLocallyConstant). Let γ be a closed curve
in C. Then:
(i) the map
is locally constant;
C\{γ} → C, z ↦→ ν(γ,z)
(ii) there exists R > 0 such that C\BR[0] ⊂ extγ.
Proof. (i): Let z0 ∈ C\{γ} and choose R > r > 0 such that BR(z0) ⊂ C\{γ}.
Consider the function
F : {γ}×Br[z0] → C, (ζ,z) ↦→ 1
ζ −z .
Then F is continuous and thus uniformly continuous. Choose δ ∈ (0,r) such that
Then
z ∈ Bδ(z0), ζ ∈ {γ} ⇒ |F(ζ,z)−F(ζ,z0)|<
π
ℓ(γ)+1 .
� � � � �
�
|ν(γ,z)−ν(γ,z0)| = �
1 1 1 �
� − dζ�
2πi γ ζ −z ζ −z0
�
≤ ℓ(γ)
2π sup|F(ζ,z)−F(ζ,z0)|
ζ∈{γ}
≤ ℓ(γ) π
2π ℓ(γ)+1
< 1
2 .
Since ν(γ,z)−ν(γ,z0) ∈ Z, this means that ν(γ,z) = ν(γ,z0).
(ii): For any z ∈ C\{γ}, we have
� �
�
|ν(γ,z)|= �
1 1
�2πi
ζ −z dζ
�
�
�
ℓ(γ) 1
� ≤
2π dist(z,{γ}) .
Since lim
|z|→∞
γ
dist(z,{γ}) = ∞, there exists R > 0 such that |ν(γ,z)|≤ ℓ(γ)
2π
1
dist(z,{γ})
for all z ∈ C such that |z|> R. Since ν(γ,z) ∈ Z for all z ∈ C\{γ}, this implies that
ν(γ,z) = 0 for all z ∈ C with |z|> R.
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