Math 411: Honours Complex Variables - University of Alberta

38 CHAPTER 5. CAUCHY’S INTEGRAL THEOREM AND FORMULA
Theorem 5.4 (Higher Derivatives). Let D ⊂ C be open, let z0 ∈ D and r > 0 be
such that Br[z0] ⊂ D, and let f: D → C be continuous such that
f(z) = 1
�
f(ζ)
2πi ζ −z dζ
∂Br(z0)
holds for all z ∈ Br(z0). Then f is infinitely often complex differentiable on Br(z0)
and satisfies
f (n) (z) = n!
�
f(ζ)
dζ
2πi ∂Br(z0) (ζ −z) n+1
(∗)
holds for all z ∈ Br(z0) and n ∈ N0.
Proof. We prove by induction on n ∈ N0: f is n-times complex differentiable and (∗)
holds.
For n = 0, the claim is clear, so suppose that it is true for some n ∈ N0. Define
F : [0,2π]×Br(z0) → C, (θ,z) ↦→ n!
2π
f(z0 +re iθ )re iθ
(z0 +re iθ −z) n+1.
Then F is continuous and, by the induction hypothesis, satisfies
f (n) (z) = n!
� � 2π
f(ζ)
dζ = F(θ,z)dθ
2πi ∂Br(z0) (ζ −z) n+1
0
for all z ∈ Br(z0). Furthermore,
∂F
∂z
0
(θ,z) = (n+1)!
2π
f(z0 +re iθ )re iθ
(z0 +re iθ −z) n+2
is continuous on [0,2π] × Br(z0). From Lemma 5.4, we thus conclude that f (n) is
holomorphic on D with
f (n+1) � 2π
�
∂F (n+1)! f(ζ)
(z) = (θ,z)dθ = dζ.
∂z 2πi ∂Br(z0) (ζ −z) n+2
Corollary 5.4.1 (Generalized Cauchy Integral Formula). Let D ⊂ C be open, and
let f: D → C be holomorphic. Then f is infinitely often complex differentiable on D.
Moreover, for any z0 ∈ D and r > 0 such that Br[z0] ⊂ D, the generalized Cauchy
integral formula
holds for all z ∈ Br(z0) and n ∈ N0.
f (n) (z) = n!
�
f(ζ)
dζ
2πi ∂Br(z0) (ζ −z) n+1