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