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 7<br />
Elementary Properties <strong>of</strong><br />
Holomorphic Functions<br />
Theorem 7.1 (Identity Theorem). Let D ⊂ C be open and connected, and let f,g:<br />
D → C be holomorphic. Then the following are equivalent:<br />
(i) f = g;<br />
(ii) the set {z ∈ D : f(z) = g(z)} has a cluster point in D;<br />
(iii) there exists z0 ∈ D such that f (n) (z0) = g (n) (z0) for all n ∈ N0.<br />
Pro<strong>of</strong>. Without loss <strong>of</strong> generality, it suffices to prove the case where g = 0.<br />
(i) =⇒ (iii) is trivial.<br />
(iii) =⇒ (ii): Let z0 ∈ D be as in (iii), and let r > 0 be such that Br(z0) ⊂ D.<br />
Then we have by Theorem 6.3 that<br />
for all z ∈ Br(z0), so that<br />
f(z) =<br />
∞�<br />
n=0<br />
f (n) (z0)<br />
(z −z0)<br />
n!<br />
n = 0<br />
Br(z0) ⊂ Z(f) := {z ∈ D : f(z) = 0}.<br />
Every point in Br(z0) is therefore a cluster point <strong>of</strong> Z(f).<br />
(ii) =⇒ (i): Let<br />
V := {z ∈ D : z is a cluster point <strong>of</strong> Z(f)},<br />
so that V �= ∅ by (ii).<br />
We claim that V is open. Let z0 ∈ V, and let r > 0 be such that Br(z0) ⊂ D.<br />
Then<br />
∞�<br />
f(z) = an(z −z0) n<br />
n=0<br />
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