1642 CHAPTER 51. FUNDAMENTALS OF COMPLEX ANALYSIS

γ1γ2

γ3

Ω

The following theorem is the general Cauchy integral formula.

Definition 51.7.18 Let {γk}nk=1 be continuous oriented curves having bounded variation.

Then this is called a cycle if whenever, z /∈ ∪nk=1γ∗k , ∑

nk=1 n(γk,z) is an integer.

By Theorem 51.7.15 if each γk is a closed curve, then {γk}nk=1 is a cycle.

Theorem 51.7.19 Let Ω be an open subset of the plane and let f : Ω→ X be analytic. Ifγk : [ak,bk]→ Ω, k = 1, · · · ,m are continuous curves having bounded variation such thatfor all z /∈ ∪m

k=1γk ([ak,bk])

m

∑k=1

n(γk,z) equals an integer

and for all z /∈Ω,m

∑k=1

n(γk,z) = 0.

Then for all z ∈Ω\∪mk=1γk ([ak,bk]) ,

f (z)m

∑k=1

n(γk,z) =m

∑k=1

12πi

∫γk

f (w)w− z

dw.

Proof: Let φ be defined on Ω×Ω by

φ (z,w)≡{ f (w)− f (z)

w−z if w ̸= zf ′ (z) if w = z

.

Then φ is analytic as a function of both z and w and is continuous in Ω×Ω. Here is why: Itis clear that d

dw φ (z, ·)(w) exists if w ̸= z. It remains to consider whether ddz φ (·,z)(z) exists.

One needs to consider

φ (z+h,z)−φ (z,z)h

=f (z+h)− f (z)

h − f ′ (z)h

We can write f (z+h) as a power series in h whenever h is suitably small.

f (z+h)− f (z)h − f ′ (z)

h=