Math 411: Honours Complex Variables - University of Alberta

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