I am confused here . isn't it ?
and also consider the residue
Use L'hospital to finish it
Here is my attempt
I'm having troubles evaluating this improper integral
Show: using residues and a closed contour.
So, i used a closed contour C that only contains the singularity . I let R be the radius of the contour and I let R tend to infinity
So,
where
I let
By Integral theorems, I get as and
By the Cauchy Residue Theorem, I get
but I can't get the answer to evaluate. I just wanna know if there is something I'm missing in complex arithmetic, or is it my set-up that is wrong
That all looks right to me. So then, when you bring everything to one side and calculate the residue, I get:
Not sure about the easiest way to evaluate that by hand, but I just reduced it to a+bi format and got as one step:
which eventually reduces down to .