51.7. THE GENERAL CAUCHY INTEGRAL FORMULA 1645

Corollary 51.7.20 Let Ω be an open set and let γk : [ak,bk]→Ω, k = 1, · · · ,m, be closed,continuous and of bounded variation. Suppose also that ∑

mk=1 n(γk,z) = 0 for all z /∈ Ω.

Then if f : Ω→ C is analytic, ∑mk=1

∫γk

f (w)dw = 0.

Proof: This follows from Theorem 51.7.19 as follows. Let

g(w) = f (w)(w− z)

where z ∈Ω\∪mk=1γk ([ak,bk]) . Then by this theorem,

0 = 0m

∑k=1

n(γk,z) = g(z)m

∑k=1

n(γk,z) =

m

∑k=1

12πi

∫γk

g(w)w− z

dw =1

2πi

m

∑k=1

∫γk

f (w)dw.

Another simple corollary to the above theorem is Cauchy’s theorem for a simply con-nected region.

Definition 51.7.21 An open set, Ω ⊆ C is a region if it is open and connected. A region,Ω is simply connected if Ĉ \Ω is connected where Ĉ is the extended complex plane. In thefuture, the term simply connected open set will be an open set which is connected and Ĉ\Ω is connected .

Corollary 51.7.22 Let γ : [a,b]→ Ω be a continuous closed curve of bounded variationwhere Ω is a simply connected region in C and let f : Ω→ X be analytic. Then∫

γ

f (w)dw = 0.

Proof: Let D denote the unbounded component of Ĉ\γ∗. Thus ∞ ∈ Ĉ\γ∗. Thenthe connected set, Ĉ \Ω is contained in D since every point of Ĉ \Ω must be in somecomponent of Ĉ\γ∗ and ∞ is contained in both Ĉ\Ω and D. Thus D must be the componentthat contains Ĉ \Ω. It follows that n(γ, ·) must be constant on Ĉ \Ω, its value being itsvalue on D. However, for z∈D,n(γ,z) = 1

2πi∫

γ1

w−z dw and so lim|z|→∞ n(γ,z) = 0 showingn(γ,z) = 0 on D. Therefore this verifies the hypothesis of Theorem 51.7.19. Let z ∈Ω∩Dand define g(w)≡ f (w)(w− z) . Thus g is analytic on Ω and by Theorem 51.7.19,

0 = n(z,γ)g(z) =1

2πi

∫γ

g(w)w− z

dw =1

2πi

∫γ

f (w)dw.

This proves the corollary.The following is a very significant result which will be used later.

Corollary 51.7.23 Suppose Ω is a simply connected open set and f : Ω→ X is analytic.Then f has a primitive, F, on Ω. Recall this means there exists F such that F ′ (z) = f (z)for all z ∈Ω.