Prove that

**Solution**

Let us consider the function

and integrate it along a quadratic counterclockwise contour with vertices where is a big odd natural number. Hence,

We note that and that .

It’s also easy to see that

Hence, as we have that

By Residue theorem we have that

It is straightforward to show that

Hence,

in the limit . The result follows.

Using the same technique with the function

we get that

An interesting result is the following

since