I must prove that .
My attempt : None, I don't know where to start. I'm really lost, I don't have any textbook but my classnotes.
I'd like a push in the right direction rather than a full answer, but everything's welcome.
I must prove that .
My attempt : None, I don't know where to start. I'm really lost, I don't have any textbook but my classnotes.
I'd like a push in the right direction rather than a full answer, but everything's welcome.
A possible solution is to integrate, using the residue theorem, the function...
... along the grey path of the figure...
... and let R tend to infinity. By Jordan's lemma is...
... and the singularities of inside the path are and . Since is an even function respect to x it will be...
Kind regards
Moltissimo grazie, ma... why do you write and not ?
Also, is equal to ?
I don't think so, as there is 2 singularities as you pointed out. I think I'm misunderstanding something.
So I should calculate where is a closed path around ? Then I sum this result to where is a closed path around . Unless I'm not getting it.
The residue theorem states that, if C is a closed path and a function f(*) has a finite number n of simple poles inside C, is...
(1)
... where...
(2)
In our example is...
(3)
... C is the gray path in figure...
... for the Jordan's lermma is...
(4)
... so that, combining (1),(2),(3) and (4), is...
(5)
In our case f(*) has only two poles inside C : , , so that is...
(6)
By (1) and (5) we have finally...
(7)
... so that...
(8)
Kind regards