Math 411: Honours Complex Variables - University of Alberta

(ii) for each compact K ⊂ D, the sequence (fn|K) ∞ n=1 converges to f|K uniformly on
K.
Proof. (i) =⇒ (ii): Let K ⊂ D be compact. For each z ∈ K, there exists a neighbourhood
Uz ⊂ D of z such that fn|Uz→ f|Uz uniformly on Uz. Since K is compact,
there exist z1,...,zm ∈ K such that
K ⊂ Uz1 ∪···∪Uzm.
Let ǫ > 0. For each j = 1,...,m, there exists nj ∈ N such that |fn(z)−f(z)|< ǫ for
all n ≥ nj and all z ∈ Uzj . Set N := max{n1,...,nm}. Then |fn(z)−f(z)|< ǫ holds
for all n ≥ N and z ∈ K.
(ii)=⇒(i): Letz0 ∈ D, andletr > 0besuchthatBr[z0] ⊂ D. SinceBr[z0]iscompact,
(fn|Br[z0]) ∞ n=1 converges uniformly on Br[z0] to f|Br[z0]. Trivially, (fn|Br(z0)) ∞ n=1
thus converges uniformly on Br(z0) to f|Br(z0).
Instead of locally uniform convergence, we therefore often speak of compact convergence.
Lemma 6.1. Let D ⊂ C be open, let γ be a curve in D, and let f,f1,f2,...: D → C
be continuous functions such that (fn|{γ}) ∞ n=1 converges to f|{γ} uniformly on {γ}.
Then we have �
γ
�
f(ζ)dζ = lim
n→∞
Proof. Let ǫ > 0, and choose N ∈ N such that
|fn(ζ)−f(ζ)|<
γ
fn(ζ)dζ.
ǫ
ℓ(γ)+1
for all n ≥ N and z ∈ {γ}. For n ≥ N, we thus obtain:
��
� �
� �
� fn � − f�
� =
��
�
� �
� (fn � −f) �
�
γ
γ
γ
≤ ℓ(γ)sup{|fn(ζ)−f(ζ)|: ζ ∈ {γ}}
≤ ǫℓ(γ)
ℓ(γ)+1
< ǫ.
Lemma 6.2. Let D ⊂ C be open, let γ be a curve in D, and let f1,f2,...: D → C be
continuous functions converging compactly to f: D → C. Then f is continuous, and
we have � �
f(ζ)dζ = lim
n→∞
fn(ζ)dζ.
γ
γ
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