Math 411: Honours Complex Variables - University of Alberta

Chapter 12
The Residue Theorem and
Applications
Definition. Let z0 ∈ C, let r > 0, and let f: Br(z0)\{z0} → C be holomorphic with
Laurent series representation
∞�
f(z) = an(z −z0) n
n=−∞
for z ∈ Br(z0) \ {z0}. Then a−1 is called the residue of f at z0 and denoted by
res(f,z0).
Remarks. 1. By Theorem 9.3, we have
for any ρ ∈ (0,r).
res(f,z0) = 1
2πi
�
∂Bρ(z0)
f(ζ)dζ
2. If f has a removable singularity at z0, then res(f,z0) = 0.
3. Suppose that f has a simple pole at z0, i.e.
∞�
f(z) = an(z −z0) n
with a−1 �= 0, then
n=−1
res(f,z0) = lim(z
−z0)f(z).
z→z0
4. Suppose that f(z) = �∞ n=−kan(z −z0) n has a pole of order k at z0. On letting
g(z) = (z−z0) kf(z), we see that res(f,z0) is the coefficient in the Taylor series
ofg(z) = �∞ n=−kan(z−z0) n+k = �∞ n=0an−k(z−z0) n corresponding to n = k−1:
res(f,z0) = g(k−1) (z0)
(k −1)! =
1 d
(k −1)!
k−1
dzk−1 �
(z −z0) k f(z) �
z=z0 .
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