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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54 CHAPTER 8. THE SINGULARITIES OF A HOLOMORPHIC FUNCTION<br />
is holomorphic and bounded and thus, by Riemann’s Removability Criterion, has a<br />
holomorphic extension h: Br(z0) → C with h(z0) = lim 1/f(z) = 0. Note that z0 is<br />
z→z0<br />
the only zero <strong>of</strong> h. Let<br />
∞�<br />
h(z) = an(z −z0) n<br />
n=0<br />
for z ∈ Br(z0) bethe power series representation <strong>of</strong> h. Set k := min{n ∈ N0 : an �= 0}.<br />
Since a0 = h(z0) = 0, we have k ≥ 1. Define<br />
˜h: Br(z0) → C, z ↦→<br />
∞�<br />
an(z −z0) n−k .<br />
Then ˜ h is holomorphic, has no zeros, and satisfies h(z) = (z−z0) k˜ h(z) for z ∈ Br(z0).<br />
For z ∈ Br(z0)\{z0}, we thus have<br />
f(z) =<br />
n=k<br />
1<br />
(z −z0) k˜ h(z) ,<br />
so that we can construct the holomorphic function<br />
�<br />
(z −z0)<br />
g: D∪{z0} C, z ↦→<br />
kf(z), z �= z0,<br />
1<br />
˜h(z0)<br />
, z = z0.<br />
Definition. Let D ⊂ C be open, let f: D → C be holomorphic, and let z0 ∈ C\D<br />
be a pole <strong>of</strong> f. Then the positive integer k in Theorem 8.2 is called the order <strong>of</strong> z0<br />
and denoted by ord(f,z0). If ord(f,z0) = 1, we call z0 a simple pole <strong>of</strong> f.<br />
Example. For m ∈ N, consider<br />
fm: C\{0} → C, z ↦→ sinz<br />
.<br />
zm We claim that f1 has a removable singularity at 0 whereas fm has a pole <strong>of</strong> order<br />
m−1 at 0 for m ≥ 2.<br />
Recall that<br />
sinz =<br />
for z ∈ C. For z �= 0, we thus have<br />
∞�<br />
n=0<br />
f1(z) = sinz<br />
z =<br />
(−1) n z 2n+1<br />
(2n+1)!<br />
∞�<br />
n=0<br />
(−1) n z 2n<br />
(2n+1)! .