....
First show
converges
Then we can write
For the integral
Write the numerator as
Then it equals to :
For the integral
then use substitution : ...
But the second integral should be zero :
After substituting it becomes
....
First show
converges
Then we can write
For the integral
Write the numerator as
Then it equals to :
For the integral
then use substitution : ...
But the second integral should be zero :
After substituting it becomes
The complex analysis offers a faster way to arrive to the result. If we apply the residue theorem we have...
(1)
... where the are the residue of the poles of with positive imaginary part. If is a pole of is [applying l'Hopital rule] ...
(2)
Now the poles of with positive imaginary part are so that is...
(3)
Kind regards