Math 411: Honours Complex Variables - University of Alberta

Definition. Let D ⊂ C be open, and let f: D → C be holomorphic. We say that f
attains w0 ∈ C with multiplicity k ∈ N at z0 ∈ D if the function
has a zero of order k at z0.
D → C, z ↦→ f(z)−w0
Theorem 13.3 (Bifurcation Theorem). Let D ⊂ C be open, let f : D → C be
holomorphic, and suppose that, at z0 ∈ D, the function f attains w0 with multiplicity
k ∈ N. Then there exist neighbourhoods V ⊂ D of z0 and W ⊂ f(V) of w0 such that,
for each w ∈ W \{w0}, there exist distinct z1,...,zk ∈ V with f(z1) = ··· = f(zk) =
w, where f attains w at each zj with multiplicity one.
Proof. In view of the Identity Theorem, we may choose ǫ > 0 with Bǫ[z0] ⊂ D such
that f(z) �= w0 and f ′ (z) �= 0 for all z in the open connected set Bǫ(z0)\{z0}.
Set V := Bǫ(z0) and γ := ∂Bǫ(z0). Choose δ > 0 such that Bδ(w0) ⊂ C\{f ◦γ},
and set W := Bδ(w0). Let w ∈ W. By the Argument Principle, the number of
times w is attained in V (counting multiplicity) is
�
1
2πi γ
f ′ �
(ζ) 1
dζ =
f(ζ)−w 2πi f◦γ
dζ
ζ −w
= ν(f ◦γ,w).
As ν(f ◦γ,·) is constant on W, the number of times w is attained in V is the same as
the number of times w0 is attained in V, i.e. k. Since f ′ (z) �= 0 for all z ∈ V \{z0}, for
w �= w0 there exist distinct z1,...,zk ∈ V \{z0} such that f(z1) = ··· = f(zk) = w;
necessarily, f attains w at each zj with multiplicity one.
Theorem 13.4 (Hurwitz’sTheorem). LetD ⊂ C be openandconnected, letf1,f2,...:
D → C be holomorphic such that (fn) ∞ n=1 converges to f compactly on D, and suppose
that Z(fn) = ∅ for n ∈ N. Then f ≡ 0 or Z(f) = ∅.
Proof. In view of Theorem 6.2, we note that f itself is holomorphic. Suppose that
f �≡ 0, but that there exists z0 ∈ Z(f). Choose ǫ > 0 such that Bǫ[z0] ⊂ D and
f(z) �= 0 for all z ∈ Bǫ[z0]\{z0}, and note that
�
1 f
0 = lim
n→∞ 2πi ∂Bǫ(z0)
′ �
n(ζ) 1 f
dζ =
fn(ζ) 2πi ∂Bǫ(z0)
′ (ζ)
dζ = ord(f,z0),
f(ζ)
which is a contradiction.
Corollary 13.4.1. Let D ⊂ C be open and connected, let f1,f2,... : D → C be
holomorphic such that (fn) ∞ n=1 converges to f compactly on D, and suppose that fn
is injective for n ∈ N. Then f is constant or injective.
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