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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Pro<strong>of</strong>. Apply Theorem 5.3 and 5.4.<br />
Example. We shall use Cauchy’s integral theorem and formula to evaluate the line<br />
integral<br />
�<br />
eζ ζ(ζ −1) dζ<br />
for various curves γ:<br />
(a) γ = ∂Bπ(−3): The function<br />
γ<br />
f: C\{1} → C, z ↦→ ez<br />
z −1<br />
is holomorphic, and we have Bπ[−3] ⊂ C\{1}. Cauchy’s integral formula thus<br />
yields:<br />
�<br />
(b) γ = ∂B1<br />
2<br />
∂Bπ(−3)<br />
eζ �<br />
f(ζ)<br />
dζ = dζ = 2πif(0) = −2πi.<br />
ζ(ζ −1) ∂Bπ(−3) ζ<br />
(i): As the integrand is holomorphic in the star-shaped domain B3(i),<br />
4<br />
Cauchy’s integral theorem yields that<br />
�<br />
∂B1 (i)<br />
2<br />
eζ dζ = 0.<br />
ζ(ζ −1)<br />
(c) γ = ∂B2(0): the method <strong>of</strong> partial fractions yields<br />
∂B2(0)<br />
1 1 1<br />
= −<br />
z(z −1) z −1 z .<br />
Since 0,1 ∈ B2(0), we obtain with the help <strong>of</strong> Cauchy’s integral formula:<br />
�<br />
eζ �<br />
dζ =<br />
ζ(ζ −1)<br />
eζ �<br />
dζ −<br />
ζ −1<br />
eζ dζ = 2πi(e−1).<br />
ζ<br />
∂B2(0)<br />
∂B2(0)<br />
Theorem 5.5 (Characterizations <strong>of</strong> Holomorphic Functions). Let D ⊂ C be open,<br />
and let f: D → C be continuous. Then the following are equivalent:<br />
(i) f is holomorphic;<br />
(ii) the Morera condition holds, i.e. �<br />
(iii) for each z0 ∈ D and r > 0 with Br[z0] ⊂ D, we have<br />
f(z) = 1<br />
�<br />
f(ζ)<br />
2πi ζ −z dζ<br />
for z ∈ Br(z0);<br />
∂∆<br />
f(ζ)dζ = 0 for each triangle ∆ ⊂ D;<br />
∂Br(z0)<br />
39