Math 411: Honours Complex Variables - University of Alberta

Definition. The function h in Theorem 9.2 is called the principal part and g is called
the secondary part of the Laurent decomposition f = g +h.
Theorem 9.3 (Laurent Coefficients). Let z0 ∈ C, let r,R ∈ [0,∞] be such that
r < R, and let f: Ar,R(z0) → C be holomorphic. Then f has a representation
f(z) =
∞�
n=−∞
an(z −z0) n
for z ∈ Ar,R(z0) as a Laurent series, which converges uniformly and absolutely on
compact subsets of Ar,R(z0). Moreover, for every n ∈ Z and ρ ∈ (r,R), the coefficients
an are uniquely determined as
an = 1
2πi
�
∂Bρ(z0)
f(ζ)
dζ.
(ζ −z0) n+1
Proof. Let g and h be as in Theorem 9.2 (in particular, with lim h(z) = 0).
|z|→∞
For z ∈ BR(z0), we have the Taylor series
g(z) =
∞�
an(z −z0) n ,
which converges uniformly and absolutely on compact subsets of BR(z0).
Define
�
˜h: A 1
0, (0) → C, z ↦→ h z0 +
r
1
�
,
z
n=0
so that ˜ h is holomorphic with lim˜h(z)
= 0. Hence,
z→0
˜ h has a removable singularity at 0
and thus extends to B1
r
(0) as a holomorphic function. This holomorphic function,
which we also denote by ˜ h, can then be expanded in a Taylor series for z ∈ B1
r
so that
˜h(z) =
h(z) =
∞�
bnz n =
n=0
∞�
bnz n ,
n=1
∞�
bn(z −z0) −n
converges uniformly and absolutely on compact subsets of C\Br[z0].
Set an := b−n for n < 0. For z ∈ Ar,R(z0), we obtain
f(z) = g(z)+h(z) =
n=1
∞�
an(z −z0) n +
n=0
∞�
a−n(z −z0) −n =
n=1
∞�
n=−∞
(0):
an(z −z0) n .
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