Math 411: Honours Complex Variables - University of Alberta

Chapter 8
The Singularities of a Holomorphic
Function
Definition. Let D ⊂ C be open, and let f : D → C be holomorphic. We call
z0 ∈ C\D anisolated singularity of f if there exists ǫ > 0 such that Bǫ(z0)\{z0} ⊂ D.
We say that the isolated singularity z0 is removable if there exists a holomorphic
function g: D ∪{z0} → C such that g|D= f.
Theorem 8.1 (Riemann’s Removability Condition). Let D ⊂ C be open, let f :
D → C be holomorphic, and let z0 ∈ C\D be an isolated singularity of f. Then the
following are equivalent:
(i) z0 is removable;
(ii) there is a continuous function g: D ∪{z0} → C such that g|D= f;
(iii) there exists ǫ > 0 with Bǫ(z0)\{z0} ⊂ D such that f is bounded on Bǫ(z0)\{z0}.
Proof. (i) =⇒ (ii) follows from the continuity of a differentiable function.
(ii) =⇒ (iii) follows from the boundedness of g on a compact set Bǫ[z0] ⊂ D.
(iii) =⇒ (i): Let C ≥ 0 be such that |f(z)|≤ C for z ∈ Bǫ(z0)\{z0}. Define
h: D ∪{z0} → C, z ↦→
� (z −z0) 2 f(z), z �= z0,
0, z = z0.
Then we have for z ∈ Bǫ(z0)\{z0} that
� �
�
�
h(z)−h(z0) �
�
� z −z0
� = |(z −z0)f(z)|≤ C|z −z0|.
Hence, h is holomorphic with h ′ (z0) = h(z0) = 0. Let h(z) = �∞ n=0an(z − z0) n
be the power series representation of h on Bǫ(z0). Then h ′ (z0) = h(z0) = 0 means
that a0 = a1 = 0, so that h(z) = �∞ n=2an(z − z0) n �
for z ∈ Bǫ(z0) and thus f(z) =
∞
n=0an+2(z −z0) n for z ∈ Bǫ(z0)\{z0}.
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