Though contour integral
I'm trying to crack this problem but I can't seem to get it right.
It's the third exercise from from chapter 2 of "Complex Analysis" by Stein & Shakarchi.
I've drawn the sector (contour C) here (sorry for the crappy MS-paint picture):
As there are no poles within the sector I set , so that leaves
Now, I assume one of these three components goes to zero as A goes to infinity and indeed the second one (with as variable) does. Then I'm stuck because what I find doesn't lead me to anything remotely like the answer Wolfram's online calculator gave me if I use integration by parts
Can anyone help me with this problem, what am I doing wrong?
I'm not familiar with complex analysis, so this probably won't be of help, but both can be done by using by-parts (twice).
I know, but I don't think I'd get away with that if this problem was in my exam.
Consider the path as you have drawn , since is analytic everywhere , we have
is the straight line starting from to
is the arc
is a straight line
Then we have
By joining the three integrals we obtain
By comparing the real part and the Im part , we have
Thank you very much, the part was the missing link. Every contour integral needs its own set of tricks it seems, very annoying, but at least I can have some peace of mind now.