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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76 CHAPTER 12. THE RESIDUE THEOREM AND APPLICATIONS<br />
Examples.<br />
� π<br />
dt<br />
1. Let a > 1. What is<br />
a+cost ?<br />
First, note that<br />
� π<br />
Let<br />
so that<br />
where<br />
0<br />
0<br />
dt<br />
a+cost<br />
� π<br />
1<br />
=<br />
2 −π<br />
dt<br />
a+cost<br />
� 2π<br />
1<br />
=<br />
2 0<br />
p(x,y) = 1 and q(x,y) = a+x,<br />
f(z) = 1<br />
iz ·<br />
=<br />
a+ 1<br />
2<br />
−i<br />
az + z2<br />
2<br />
1<br />
+ 1<br />
2<br />
� z + 1<br />
z<br />
�<br />
−2i<br />
=<br />
z2 +2az +1<br />
−2i<br />
=<br />
(z −z1)(z −z2) ,<br />
dt<br />
a+cost .<br />
z1 = −a+ √ a 2 −1 ∈ D and z2 = −a− √ a 2 −1 /∈ D, (12.1)<br />
on noting that 1+ √ a 2 −1 > a implies that z1 > −1.<br />
By Proposition 12.1, we thus obtain<br />
� π � 2π<br />
dt 1 dt<br />
=<br />
0 a+cost 2 0 a+cost<br />
= πi res(f,z1)<br />
= πi· −2i<br />
z1 −z2<br />
2π<br />
=<br />
2 √ a2 −1<br />
π<br />
= √ .<br />
a2 −1<br />
2. Let a > 0. What is<br />
Let<br />
� 2π<br />
0<br />
dt<br />
(a+cost) 2?<br />
p(x,y) = 1 and q(x,y) = (a+x) 2 ,