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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30 CHAPTER 5. CAUCHY’S INTEGRAL THEOREM AND FORMULA<br />
with<br />
ℓ(∂∆n) = 1<br />
2nℓ(∂∆) and �� ���<br />
� ��<br />
� �<br />
f�<br />
� ≤ 4n�<br />
�<br />
�<br />
�<br />
f�<br />
�<br />
for n ∈ N.<br />
Let z0 ∈ � ∞<br />
n=1 ∆n, and define<br />
∂∆<br />
∂∆n<br />
r: D → C, z ↦→ f(z)−f(z0)−f ′ (z0)(z −z0),<br />
|r(z)| � �<br />
so that lim = 0 and (r −f) =<br />
z→z0 |z −z0| γ γ [−f(z0)−f ′ (z0)(z −z0)]dz = 0 for each<br />
closed curve γ in D (noting that the integrand has an antiderivative). Consequently,<br />
��<br />
�<br />
�<br />
�<br />
� ��<br />
� �<br />
f�<br />
� ≤ 4n�<br />
�<br />
�<br />
�<br />
r�<br />
�<br />
∂∆<br />
∂∆n<br />
for n ∈ N. Let ǫ > 0 and choose δ > 0 such that<br />
� �<br />
�<br />
�<br />
r(z) �<br />
�<br />
�z<br />
−z0<br />
� ≤<br />
ǫ<br />
[ℓ(∂∆)] 2<br />
for all z ∈ D with |z −z0|< δ. Choose n ∈ N such that ∆n ⊂ Bδ(z0). For z ∈ ∆n,<br />
this means that<br />
|z −z0|≤ ℓ(∂∆n) = 1<br />
2nℓ(∂∆). We thus obtain:<br />
���� � � ��<br />
� �<br />
f�<br />
� ≤ 4n�<br />
�<br />
�<br />
�<br />
r�<br />
�<br />
∂∆<br />
∂∆n<br />
≤ 4 n ℓ(∂∆n) sup<br />
ζ∈∂∆n<br />
≤ 2 n ℓ(∂∆) sup<br />
ζ∈∂∆n<br />
≤ ǫ.<br />
As ǫ > 0 was arbitrary, this proves the claim.<br />
|r(ζ)|= 2 n ℓ(∂∆) sup<br />
ǫ<br />
|ζ −z0|<br />
[ℓ(∂∆)] 2<br />
� �� �<br />
≤ 1<br />
2 nℓ(∂∆)<br />
ζ∈∂∆n<br />
|r(ζ)|<br />
Definition. A set D ⊂ C is called star shaped if there exists z0 ∈ D such that<br />
[z0,z] ⊂ D for each z ∈ D. The point z0 is called a center for D.<br />
Theorem 5.2. Let D ⊂ C be open and star shaped with center z0, and let f: D → C<br />
be continuous such that �<br />
f(ζ)dζ = 0<br />
∂∆<br />
for each triangle ∆ ⊂ D with z0 as a vertex. Then f has an antiderivative.