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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36 CHAPTER 5. CAUCHY’S INTEGRAL THEOREM AND FORMULA<br />
Then g is continuous on D and holomorphic on D \ {z}. By Lemma 5.1, g has an<br />
antiderivative on BR(z0). Since ∂Br(z0) is closed, this means that<br />
�<br />
0 =<br />
∂Br(z0)<br />
using the identity �<br />
yields the claim.<br />
�<br />
g(ζ)dζ =<br />
∂Br(z0)<br />
∂Br(z0)<br />
�<br />
f(ζ) 1<br />
dζ −f(z)<br />
ζ −z ∂Br(z0) ζ −z dζ,<br />
� �� �<br />
=2πi<br />
1 dζ = 2πi established by Lemma 5.2. Division by 2πi<br />
ζ−z<br />
Corollary 5.3.1 (Mean Value Equation). Let D ⊂ C be open, let f : D → C be<br />
holomorphic, and let z0 ∈ D and r > 0 be such that Br[z0] ⊂ D. Then we have<br />
Pro<strong>of</strong>. Parametrize ∂Br(z0) as<br />
The Cauchy Integral Formula then yields<br />
f(z0) = 1<br />
� 2π<br />
f(z0 +re<br />
2π 0<br />
iθ )dθ.<br />
γ: [0,2π] → C, θ ↦→ z0 +re iθ .<br />
f(z0) = 1<br />
�<br />
2πi<br />
= 1<br />
2πi<br />
= 1<br />
2π<br />
γ<br />
� 2π<br />
0<br />
� 2π<br />
0<br />
f(ζ)<br />
dζ<br />
ζ −z0<br />
f(z0 +re iθ )<br />
re iθ ire iθ dθ<br />
f(z0 +re iθ )dθ.<br />
Lemma 5.3. Let D ⊂ RN be open, and let f: [a,b]×D → R be continuous such that<br />
: [a,b]×D → R all exist and are continuous. Define<br />
∂f ∂f<br />
,..., ∂x1 ∂xN<br />
� b<br />
g: D → R, x ↦→ f(t,x)dt.<br />
Then g is continuously partially differentiable with<br />
for x ∈ D and j = 1,...,N.<br />
� b<br />
∂g ∂f<br />
(x) = (t,x)dt<br />
∂xj ∂xj<br />
a<br />
a