Math 411: Honours Complex Variables - University of Alberta

Examples.
1. Let
f(z) = eiz
z 2 +1 ,
so that f has a simple pole at z0 = i. It follows that
2. Let
res(f,i) = lim(z
−i)f(z) = lim
z→i z→i
f(z) = cos(πz)
sin(πz) ,
e iz
z +i
= − i
2e .
so that f has a simple pole at each n ∈ Z. For n ∈ Z, we thus have:
3. Let
res(f,n) = lim(z
−n)
z→n cos(πz)
sin(πz)
cos(πz)
= lim(z
−n)
z→n
= 1
π lim
z→n
= 1
π .
f(z) =
sin(πz)−sin(πn)
πz −πn
sin(πz)−sin(πn) cos(πz)
1
(z 2 +1) 3;
then f has a pole of order 3 at z0 = i. With
we have
so that
g(z) = (z −i) 3 f(z) =
g ′ (z) = − 3
(z +i) 4
1
(z +i) 3,
and g ′′ (z) = 12
(z +i) 5,
res(f,i) = 1 12
= −3i
2(2i)
5 16 .
Theorem 12.1 (Residue Theorem). Let D ⊂ C be open and simply connected,
z1,...,zn ∈ D be such that zj �= zk for j �= k, f : D \ {z1,...,zn} → C be holomorphic,
and γ be a closed curve in D \{z1,...,zn}. Then we have
�
n�
f(ζ)dζ = 2πi ν(γ,zj) res(f,zj).
γ
j=1
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