1634 CHAPTER 51. FUNDAMENTALS OF COMPLEX ANALYSIS

Then let z ∈ ∩∞k=1Tk and note that by assumption, f ′ (z) exists. Therefore, for all k large

enough, ∫∂Tk

f (w)dw =∫

∂Tk

f (z)+ f ′ (z)(w− z)+g(w)dw

where ||g(w)|| < ε |w− z| . Now observe that w→ f (z) + f ′ (z)(w− z) has a primitive,namely,

F (w) = f (z)w+ f ′ (z)(w− z)2 /2.

Therefore, by Corollary 50.0.14.∫∂Tk

f (w)dw =∫

∂Tk

g(w)dw.

From the definition, of the integral,

α

4k ≤∣∣∣∣∣∣∣∣∫

∂Tk

g(w)dw∣∣∣∣∣∣∣∣≤ ε diam(Tk)(length of ∂Tk)

≤ ε2−k (length of T )diam(T )2−k,

and soα ≤ ε (length of T )diam(T ) .

Since ε is arbitrary, this shows α = 0, a contradiction. Thus∫

∂T f (w)dw = 0 as claimed.This fundamental result yields the following important theorem.

Theorem 51.7.2 (Morera1) Let Ω be an open set and let f ′ (z) exist for all z ∈ Ω. LetD≡ B(z0,r)⊆Ω. Then there exists ε > 0 such that f has a primitive on B(z0,r+ ε).

Proof: Choose ε > 0 small enough that B(z0,r+ ε) ⊆ Ω. Then for w ∈ B(z0,r+ ε) ,define

F (w)≡∫

γ(z0,w)f (u)du.

Then by the Cauchy Goursat theorem, and w ∈ B(z0,r+ ε) , it follows that for |h| smallenough,

F (w+h)−F (w)h

=1h

∫γ(w,w+h)

f (u)du

=1h

∫ 1

0f (w+ th)hdt =

∫ 1

0f (w+ th)dt

which converges to f (w) due to the continuity of f at w. This proves the theorem.The following is a slight generalization of the above theorem which is also referred to

as Morera’s theorem.

1Giancinto Morera 1856-1909. This theorem or one like it dates from around 1886