Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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Chapter 6<br />
Convergence <strong>of</strong> Holomorphic<br />
Functions<br />
Recall the following definition:<br />
Definition. Let D ⊂ C be open. A sequence (fn) ∞ n=1 <strong>of</strong> C-valued functions on D is<br />
said to converge uniformly on D to f: D → C if, for each ǫ > 0, there exists N ∈ N<br />
such that |fn(z)−f(z)|< ǫ for all n ≥ N and all z ∈ D.<br />
We recall the following theorem from analysis:<br />
Theorem 6.1 (Uniform Convergence Preserves Continuity). Let D ⊂ C be open, and<br />
let (fn) ∞ n=1 be a sequence <strong>of</strong> continuous, C-valued functions on D converging uniformly<br />
on D to f: D → C. Then f is continuous.<br />
We now “localize” the notion <strong>of</strong> uniform convergence:<br />
Definition. Let D ⊂ C be open. Then a sequence (fn) ∞ n=1 <strong>of</strong> C-valued functions on<br />
D is said to converge locally uniformly on D to f : D → C if, for each z0 ∈ D, there<br />
exists a neighbourhood U ⊂ D <strong>of</strong> z0 such that (fn|U) ∞ n=1 converges to f|U uniformly<br />
on U.<br />
Proposition 6.1 (Local Uniform Convergence). Let D ⊂ C be open, and let (fn) ∞ n=1<br />
be a sequence <strong>of</strong> continuous, C-valued functions on D converging locally uniformly on<br />
D to f: D → C. Then f is continuous.<br />
Pro<strong>of</strong>. Let z0 ∈ D, and let U ⊂ D be a neighbourhood <strong>of</strong> z0 such that fn|U→ f|U<br />
uniformly on U By Theorem 6.1, f|U is continuous. Hence, f is continuous at z0.<br />
Proposition 6.2 (Compact Convergence). Let D ⊂ C be open, and let f,f1,f2,...:<br />
D → C be functions. Then the following are equivalent:<br />
(i) (fn) ∞ n=1 converges to f locally uniformly on D;<br />
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