Math 411: Honours Complex Variables - University of Alberta

Chapter 6
Convergence of Holomorphic
Functions
Recall the following definition:
Definition. Let D ⊂ C be open. A sequence (fn) ∞ n=1 of C-valued functions on D is
said to converge uniformly on D to f: D → C if, for each ǫ > 0, there exists N ∈ N
such that |fn(z)−f(z)|< ǫ for all n ≥ N and all z ∈ D.
We recall the following theorem from analysis:
Theorem 6.1 (Uniform Convergence Preserves Continuity). Let D ⊂ C be open, and
let (fn) ∞ n=1 be a sequence of continuous, C-valued functions on D converging uniformly
on D to f: D → C. Then f is continuous.
We now “localize” the notion of uniform convergence:
Definition. Let D ⊂ C be open. Then a sequence (fn) ∞ n=1 of C-valued functions on
D is said to converge locally uniformly on D to f : D → C if, for each z0 ∈ D, there
exists a neighbourhood U ⊂ D of z0 such that (fn|U) ∞ n=1 converges to f|U uniformly
on U.
Proposition 6.1 (Local Uniform Convergence). Let D ⊂ C be open, and let (fn) ∞ n=1
be a sequence of continuous, C-valued functions on D converging locally uniformly on
D to f: D → C. Then f is continuous.
Proof. Let z0 ∈ D, and let U ⊂ D be a neighbourhood of z0 such that fn|U→ f|U
uniformly on U By Theorem 6.1, f|U is continuous. Hence, f is continuous at z0.
Proposition 6.2 (Compact Convergence). Let D ⊂ C be open, and let f,f1,f2,...:
D → C be functions. Then the following are equivalent:
(i) (fn) ∞ n=1 converges to f locally uniformly on D;
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