Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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38 CHAPTER 5. CAUCHY’S INTEGRAL THEOREM AND FORMULA<br />
Theorem 5.4 (Higher Derivatives). Let D ⊂ C be open, let z0 ∈ D and r > 0 be<br />
such that Br[z0] ⊂ D, and let f: D → C be continuous such that<br />
f(z) = 1<br />
�<br />
f(ζ)<br />
2πi ζ −z dζ<br />
∂Br(z0)<br />
holds for all z ∈ Br(z0). Then f is infinitely <strong>of</strong>ten complex differentiable on Br(z0)<br />
and satisfies<br />
f (n) (z) = n!<br />
�<br />
f(ζ)<br />
dζ<br />
2πi ∂Br(z0) (ζ −z) n+1<br />
(∗)<br />
holds for all z ∈ Br(z0) and n ∈ N0.<br />
Pro<strong>of</strong>. We prove by induction on n ∈ N0: f is n-times complex differentiable and (∗)<br />
holds.<br />
For n = 0, the claim is clear, so suppose that it is true for some n ∈ N0. Define<br />
F : [0,2π]×Br(z0) → C, (θ,z) ↦→ n!<br />
2π<br />
f(z0 +re iθ )re iθ<br />
(z0 +re iθ −z) n+1.<br />
Then F is continuous and, by the induction hypothesis, satisfies<br />
f (n) (z) = n!<br />
� � 2π<br />
f(ζ)<br />
dζ = F(θ,z)dθ<br />
2πi ∂Br(z0) (ζ −z) n+1<br />
0<br />
for all z ∈ Br(z0). Furthermore,<br />
∂F<br />
∂z<br />
0<br />
(θ,z) = (n+1)!<br />
2π<br />
f(z0 +re iθ )re iθ<br />
(z0 +re iθ −z) n+2<br />
is continuous on [0,2π] × Br(z0). From Lemma 5.4, we thus conclude that f (n) is<br />
holomorphic on D with<br />
f (n+1) � 2π<br />
�<br />
∂F (n+1)! f(ζ)<br />
(z) = (θ,z)dθ = dζ.<br />
∂z 2πi ∂Br(z0) (ζ −z) n+2<br />
Corollary 5.4.1 (Generalized Cauchy Integral Formula). Let D ⊂ C be open, and<br />
let f: D → C be holomorphic. Then f is infinitely <strong>of</strong>ten complex differentiable on D.<br />
Moreover, for any z0 ∈ D and r > 0 such that Br[z0] ⊂ D, the generalized Cauchy<br />
integral formula<br />
holds for all z ∈ Br(z0) and n ∈ N0.<br />
f (n) (z) = n!<br />
�<br />
f(ζ)<br />
dζ<br />
2πi ∂Br(z0) (ζ −z) n+1