Math 411: Honours Complex Variables - University of Alberta

That is, F ′ (z) = f(z).
Theorem 4.1 (Antiderivative Theorem). Let D ⊂ C be open and connected and let
f: D → C be continuous. Then the following are equivalent:
(i) f has an antiderivative;
(ii) �
f(ζ)dζ = 0 for any closed, piecewise smooth curve γ in D;
γ
(iii) for any piecewise smooth curve γ in D, the value of �
f depends only on the
γ
inital point and the endpoint of γ.
Proof. (i) =⇒ (ii) is Proposition 4.1.
(ii) =⇒ (iii): Let γ,Γ: [a,b] → D be piecewise smooth curves with γ(a) = Γ(a)
and γ(b) = Γ(b). Then γ ⊕Γ− is a closed, piecewise smooth curve, so that
� � � � �
0 = f = f + f = f − f.
γ⊕Γ −
γ
(iii) =⇒ (i): Fix z0 ∈ D. For each z ∈ D, choose a piecewise smooth curve
γz : [a,b] → D with γz(a) = z0 and γz(b) = z and let
�
F: D → C, z ↦→ f(ζ)dζ.
For each z ∈ D, choose δ > 0 such that Bδ(z) ⊂ D and note for w ∈ Bδ(z) that
�
F(w) = f(ζ)dζ
γw �
= f(ζ)dζ by (iii)
γz⊕[z,w]
� �
= f(ζ)dζ + f(ζ)dζ
γz
�
[z,w]
= F(z)+ f(ζ)dζ.
Γ −
[z,w]
Lemma 4.1 then implies that F is an antiderivative for f.
Fromnowon, weshall use theword curveasshorthandforpiecewise smooth curve.
γz
γ
Γ
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