Math 411: Honours Complex Variables - University of Alberta

80 CHAPTER 12. THE RESIDUE THEOREM AND APPLICATIONS
2. What is
� ∞
−∞
1
(x2 dx, where n ∈ N?
+1) n
The polynomial q(z) := (z 2 +1) n has zeros of order n at ±i. Define
so that
and thus
It follows that � ∞
−∞
in particular, we have
� ∞
−∞
n 1
g(z) = (z −i)
q(z) = (z +i)−n ,
g (n−1) (z) = (−n)···(−2n+2)(z +i) −2n+1
�
1
res
q ,i
�
= g(n−1) (i)
(n−1)!
1
=
(n−1)!2 2n−1i ·n···(2n−2)
=
(2n−2)!
i2 2n−1 (n−1)! 2.
1
(x2 �
1
dx = 2πi res
+1) n q ,i
�
= π
22n−2 (2n−2)!
(n−1)! 2;
1
x2 dx = π,
+1
� ∞
1
(x2 π
dx = , and
+1) 2 2
−∞
Problem 12.3.
(a) Prove that sinθ ≥ 2 π
θ for 0 ≤ θ ≤
π 2 .
(b) Use part (a) to show that for R > 0 that
� π
e −Rsinθ dθ < π
R .
0
� ∞
−∞
1
(x2 3π
dx =
+1) 3 8 .
(c) Let CR be the semicircular contour {Reiθ : 0 ≤ θ ≤ π}, with R > 0. Use part (b)
to establish Jordan’s Lemma: �� ���
e iz �
�
dz�
� < π.
Problem 12.4.
CR
Let D be an open set. If f: D\{z0} → C is holomorphic, where f has a simple pole
at z0, and Cr = {z0 +reiθ : α ≤ θ ≤ β}, prove the Fractional Residue Theorem:
�
lim f(z)dz = (β −α)ires(f,z0).
r→0
Cr