1688 CHAPTER 53. RESIDUES
x
y
Let γr (t) = reit , t ∈ [0,π] and let σ r (t) = t : t ∈ [−r,r] . Thus γr parameterizes the topcurve and σ r parameterizes the straight line from −r to r along the x axis. Denoting by Γrthe closed curve traced out by these two, we see from simple estimates that
limr→∞
∫γr
11+ z4 dz = 0.
This follows from the following estimate.∣∣∣∣∫γr
11+ z4 dz
∣∣∣∣≤ 1r4−1
πr.
Therefore, ∫∞
−∞
11+ x4 dx = lim
r→∞
∫Γr
11+ z4 dz.
We compute∫
Γr1
1+z4 dz using the method of residues. The only residues of the integrandare located at points, z where 1+ z4 = 0. These points are
z = −12
√2− 1
2i√
2,z =12
√2− 1
2i√
2,
z =12
√2+
12
i√
2,z =−12
√2+
12
i√
2
and it is only the last two which are found in the inside of Γr. Therefore, we need tocalculate the residues at these points. Clearly this function has a pole of order one at eachof these points and so we may calculate the residue at α in this list by evaluating
limz→α
(z−α)1
1+ z4
Thus
Res(
f ,12
√2+
12
i√
2)
= limz→ 1
2√
2+ 12 i√
2
(z−(
12
√2+
12
i√
2))
11+ z4
= −18
√2− 1
8i√
2