1624 CHAPTER 51. FUNDAMENTALS OF COMPLEX ANALYSIS

Now for all h sufficiently small, there exists a constant C independent of such h such that∣∣∣∣ 1(−w+ z+h)(−w+ z)

− 1(−w+ z)(−w+ z)

∣∣∣∣=

∣∣∣∣∣ h

(w− z−h)(w− z)2

∣∣∣∣∣≤C |h|

and so, the integrand converges uniformly as h→ 0 to

=f (w)

(w− z)2

Therefore, the limit as h→ 0 may be taken inside the integral to obtain

f ′ (z) =1

2πi

∫γ

f (w)

(w− z)2 dw.

Continuing in this way, yields 51.3.10.This is a very remarkable result. It shows the existence of one continuous derivative im-

plies the existence of all derivatives, in contrast to the theory of functions of a real variable.Actually, more than what is stated in the theorem was shown. The above proof establishesthe following corollary.

Corollary 51.3.12 Suppose f is continuous on ∂B(z0,r) and suppose that for all z ∈B(z0,r) ,

f (z) =1

2πi

∫γ

f (w)w− z

dw,

where γ (t) ≡ z0 + reit , t ∈ [0,2π] . Then f is analytic on B(z0,r) and in fact has infinitelymany derivatives on B(z0,r) .

Another application is the following lemma.

Lemma 51.3.13 Let γ (t) = z0+reit , for t ∈ [0,2π], suppose fn→ f uniformly on B(z0,r),and suppose

fn (z) =1

2πi

∫γ

fn (w)w− z

dw (51.3.11)

for z ∈ B(z0,r) . Then

f (z) =1

2πi

∫γ

f (w)w− z

dw, (51.3.12)

implying that f is analytic on B(z0,r) .

Proof: From 51.3.11 and the uniform convergence of fn to f on γ ([0,2π]) , the integralsin 51.3.11 converge to

12πi

∫γ

f (w)w− z

dw.