Math 411: Honours Complex Variables - University of Alberta

Chapter 11
The General Cauchy Integral
Theorem
Definition. Let D ⊂ C be open. We call a closed curve γ in D homologous to zero
if ν(γ,z) = 0 for each z ∈ C\D. That is, the interior of γ is a subset of D.
Definition. An open connected subset D of C is simply connected if every closed
curve in D is homologous to zero. Equivalently, the interior of every closed curve in
D is a subset of D.
Theorem 11.1 (Cauchy’s Integral Formula). Let D ⊂ C be open, let f : D → C be
holomorphic, and let γ be a closed curve in D that is homologous to zero. Then, for
n ∈ N0 and z ∈ D \{γ}, we have
ν(γ,z)f (n) (z) = n!
�
2πi γ
f(ζ)
dζ.
(ζ −z) n+1
Proof. It is enough to prove the claim for n = 0: for n ≥ 1, differentiate the integral
with respect to z and use induction.
Define
g: D ×D → C, (w,z) ↦→
� f(w)−f(z)
w−z , w �= z,
f ′ (z), w = z.
We claim that g is continuous. To see this, let (w0,z0) ∈ D × D. As g is clearly
continuous at (w0,z0) if w0 �= z0, we need only show that g is continuous at (z0,z0).
Given ǫ > 0, choose δ > 0 small enough that Bδ[z0] ⊂ D and |f ′ (z) −f ′ (z0)|< ǫ for
all z ∈ Bδ[z0]. For (w,z) ∈ Bδ(z0)×Bδ(z0) we find:
• if w = z:
|g(w,z)−g(z0,z0)|= |f ′ (z)−f ′ (z0)|< ǫ;
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