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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12.1. APPLICATIONS OF THE RESIDUE THEOREM TO REAL INTEGRALS79<br />
We then find that<br />
� ∞<br />
Examples.<br />
1. What is<br />
� ∞<br />
0<br />
−∞<br />
�<br />
p(x)<br />
dx = lim<br />
q(x) r→∞<br />
1<br />
dx?<br />
1+x 6<br />
�<br />
= lim<br />
r→∞<br />
�<br />
= lim<br />
r→∞<br />
[−r,r]<br />
[−r,r]<br />
p(ζ)<br />
q(ζ) dζ<br />
p(ζ)<br />
dζ + lim<br />
q(ζ) r→∞<br />
p(ζ)<br />
[−r,r]⊕γr q(ζ) dζ<br />
= 2πi �<br />
�<br />
p<br />
res<br />
q ,z<br />
�<br />
.<br />
z∈H<br />
�<br />
γr<br />
p(ζ)<br />
q(ζ) dζ<br />
The zeros <strong>of</strong> q(z) = 1 + z6 are <strong>of</strong> the form eiθ where θ ∈ [0,2π) is such that<br />
ei6θ = −1 = eiπ , i.e. 6θ − π ∈ 2πZ, so that θ = π π 5π 7π 3π 11π , , , , , . For<br />
6 2 6 6 2 6<br />
k = 1,...,6, let<br />
zk = e i(2k−1)π 6.<br />
Then 1<br />
q has a simple pole at zk for k = 1,...,6.<br />
By Problem 12.1, we have<br />
res<br />
so that by, Proposition 12.2,<br />
� ∞<br />
0<br />
1 1<br />
dx =<br />
1+x 6 2<br />
�<br />
1<br />
q ,zk<br />
�<br />
� ∞<br />
= 1<br />
q ′ 1<br />
=<br />
(zk) 6z5 k<br />
1<br />
dx<br />
−∞ 1+x 6<br />
3�<br />
�<br />
1<br />
= πi res<br />
q<br />
k=1<br />
,zk<br />
�<br />
= − πi<br />
�<br />
6<br />
= − πi<br />
�<br />
cos<br />
6<br />
π π<br />
+isin<br />
6 6<br />
= π<br />
�<br />
2sin<br />
6<br />
π<br />
6 +1<br />
�<br />
= π<br />
3 .<br />
e iπ<br />
6 +e iπ<br />
2 +e i5π<br />
6<br />
�<br />
= − zk<br />
6 ,<br />
+i+cos 5π<br />
6<br />
�<br />
5π<br />
+isin<br />
6