Math 411: Honours Complex Variables - University of Alberta

For the converse, let g: Br(z0) → C be holomorphic such that g(z0) �= 0 and
f(z) = g(z)
(z−z0) k for z ∈ Br(z0)\{z0}. Let g(z) = �∞ n=0bn(z−z0) n for z ∈ Br(z0)
be the Taylor series of g, so that
f(z) =
∞�
bn(z −z0) n−k
n=0
for z ∈ Br(z0)\{z0}. The uniqueness of the Laurent representation yields that
an = bn+k for n ≥ −k and an = 0 for n < −k.
(iii) This follows from (i) and (ii) by elimination.
Examples.
1. Let
Then f has the Laurent representation
1
−
f: C\{0} → C, z ↦→ e z2 .
f(z) =
∞� (−1) n 1
n! z2n n=0
for z ∈ C\{0} and thus has an essential singularity at 0.
2. Let
so that
f: C\{0} → C, z ↦→ ez −1
z3 ,
f(z) = 1
z 3
∞�
n=1
z n
n! =
∞� zn−3 n=1
for z ∈ C\{0}. Hence, f has a pole of order two at 0.
Remark. TheLaurentrepresentationofaholomorphicfunctiononanannulusAr,R(z0)
depends not only on z0, but also on r and R.
Example. Consider the function
f: C\{1,3} → C, z ↦→
and note that
f(z) = 1 1
−
1−z 3−z .
Then f has the following Laurent representations:
n!
2
z 2 −4z +3 ,
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