Derive the integration formula
By using the residue theorem, I integrated the function from to , I can only obtain the result , which means my answer is .
I couldnt find my mistake.Thank you.
Another way: denote , then . Using the substitution we get (Dirichlet's integral) . So, . For we get .
Then,
@fareastmovement: Show your complete work by the method of residues (of course if you want) and we can check it.
Edited: Method not valid, see the posts below.
Fubini's theorem is not satsified. So to justify changing the order of integration, let's write the double integral more formally as a limit.
Since the Sine Integral is bounded above by 2, we can use the Dominated Convergence Theorem to justify bringing the limit inside of the integral.