35.7. LIOUVILLE’S THEOREM 717

This formula shows the famous two identities,

eiπ =−1 and e2πi = 1.

This properties of zeros of an analytic function can be used to verify with no effort thatidentities which hold for z real continue to hold for z complex and this can be done with noeffort.

35.7 Liouville’s TheoremNow the following is the general Cauchy integral formula.

Theorem 35.7.1 Let U along with its boundary Γ satisfy satisfy Green’s theorem and letf be analytic on an open set V containing U ∪Γ and let γ be an orientation of Γ such thatGreen’s theorem holds. Thus,

n(γ,z)≡ 12πi

∫γ

1w− z

dw = 1

Then if z ∈U,

f (z) =1

2πi

∫γ

f (w)w− z

dw

Proof: Consider the function

g(w)≡

{f (w)− f (z)

w−z if w ̸= zf ′ (z) if w = z

(35.17)

It remains to consider whether g′ (z) exists for z ∈ V . Then from the Theorem 35.6.3, wecan write f (z+h) as a power series in h whenever h is suitably small.

f (z+h)− f (z)h − f ′ (z)

h=

1h

(1h

(f ′ (z)h+

12!

f ′′ (z)h2 +13!

f ′′′ (z)h3 + · · ·)− f ′ (z)

)=

1h

((f ′ (z)+

12!

f ′′ (z)h+13!

f ′′′ (z)h2 + · · ·)− f ′ (z)

)=

12!

f ′′ (z)+13!

f ′′′ (z)h+ higher order terms

Thus the limit of the difference quotient exists and is 12! f ′′ (z). It follows that

0 =1

2πi

∫γ

g(w)dw =1

2πi

∫γ

f (w)w− z

dw− 12πi

∫γ

f (z)w− z

dw

=1

2πi

∫γ

f (w)w− z

dw− f (z) ■

The following is a spectacular application. It is Liouville’s theorem.