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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γ1<br />
Then it is clear that<br />
� �<br />
1<br />
dζ +<br />
ζ −z<br />
γ1<br />
γ2<br />
z0<br />
z<br />
�<br />
1<br />
dζ =<br />
ζ −z ∂Br(z0)<br />
z0<br />
z<br />
�<br />
1<br />
dζ +<br />
ζ −z ∂Bǫ(z) −<br />
γ2<br />
1<br />
ζ −z dζ.<br />
The sketches also show that there exist star-shaped open set Dj ⊂ C \ {z} with<br />
{γj} ⊂ Dj for j = 1,2. Cauchy’s integral theorem for star-shaped domains thus<br />
yields that<br />
for j = 1,2, which proves the claim.<br />
�<br />
γj<br />
1<br />
dζ = 0<br />
ζ −z<br />
Theorem 5.3 (Cauchy’s Integral Formula for Circles). Let D ⊂ C be open, let<br />
f: D → C be holomorphic, and let z0 ∈ D and r > 0 be such that Br[z0] ⊂ D. Then<br />
we have<br />
for all z ∈ Br(z0).<br />
f(z) = 1<br />
�<br />
f(ζ)<br />
2πi ∂Br(z0) ζ −z dζ<br />
Remark. A consequence <strong>of</strong> Theorem 5.3 is that the values <strong>of</strong> f on all <strong>of</strong> Br[z0] are<br />
pre-determined by those on ∂Br(z0)!<br />
Pro<strong>of</strong> <strong>of</strong> Theorem 5.3. Since Br[z0] is compact, we may choose R > 0 be such that<br />
Br[z0] ⊂ BR(z0) ⊂ D. Let z ∈ Br(z0), and note that z is a center for the star-shaped<br />
domain BR(z0).<br />
Define<br />
g: D → C, ζ ↦→<br />
� f(ζ)−f(z)<br />
ζ−z , ζ �= z,<br />
f ′ (z), ζ = z.<br />
35