Math 411: Honours Complex Variables - University of Alberta

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