Use a theorem to show that:
I was moving threads and found this one.
Let me try to find this integral.
Here we need to use two theorems.
Michael Jordan's Lemma: Let be a meromorphic function in the upper half-plane with finitely many poles: in upper half-plane. Given that in the upper-half plane. Then for each positive number we have:
where
Lemma: Let has a simple pole at with residue . Let be the curve .
Then,
Now we can evaluate,
Consider the meromorphic function,
The bad thing is that we cannot use Jordan's Lemma because the poles of this function are . We disregard because we never will be working in the lower half plane, however lies on the real axis! So the conditions of Jordan's Lemma are not fullfiled. In order to avoid that we will use the other lemma by drawing a small circular contor around is "bad" point. Look at picture below.
The bottom is divided into three parts: which is the left-line segment moving from to . Also, which is the semi-circular contour (notice, this is negatively oriented). Finally, which is moving from to .
Now the only pole in upper half plane of is . And
Now by this generalization of Michael Jordan's Lemma we have (notice the negative):
Now if we take the limit at by that Lemma we have:
Since
We have,
Thus,
Thus,
And,