Math 411: Honours Complex Variables - University of Alberta

Chapter 13
Function Theoretic Consequences
of the Residue Theorem
Definition. Let D ⊂ C be open. We call S ⊂ D discrete in D if it has no cluster
points in D.
Example. If D is open and connected, and f: D → C is holomorphic and not identically
zero, then Z(f) is discrete.
Remark. Let S ⊂ D be discrete, and let K ⊂ D be compact. If K ∩S were infinite,
then K ∩S would have cluster points, which would lie in K ⊂ D. Thus, K ∩S must
be finite.
Remark. LetS beadiscretesubset ofanopensetD. Forsuitablysmallradiir(z) > 0,
we note that D can be expressed as a countable union of compact sets:
D = �
Br(z)[z].
z∈D∩Q 2
On denoting these compact sets as {Kn} ∞ n=1 , we see that each Kn ∩S is finite. Thus
S =
is either finite or countably infinite.
∞�
(Kn ∩S).
n=1
Example. If D is open and connected, and f: D → C is holomorphic and not identically
zero, then Z(f) is at most countably infinite.
Proposition 13.1. Let D ⊂ C be open, let γ be a closed curve in D, and let S ⊂ D
be discrete. Then S ∩intγ is finite.
Proof. By Proposition 10.2(ii), there exists R > 0 such that intγ ⊂ BR[0].
Definition. Denote the set of poles of a holomorphic function f by P(f).
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