I am trying to confirm the following integral using the residue theorem but I'm stuck in the end and I don't know what I did wrong:
Here's my solution:
We chose \gamma to be the positively oriented semicircle around the origin of radius R, so we get:
and we are going to apply the residue thm on the following holomorphic function:
where and c_k are the zeroes of coshz, i.e.
First, we show that the integral over the semicircle vanishes for big R:
Using the estimation lemma, where ||-|| denotes the supremum norm, we get
so it follows that
using l'Hospitals rule.
so we got: for R very big, and
Now all we have to do is compute the residue of f in its poles. Since they're all poles of order 1 we proceed:
so, finally we obtain:
but the series is DIVERGENT?!
Now, this means that whether I did something wrong using the residue thm., or I computed wrong the residues.
What is interesting to me is that the residue sum here appears to be actually a residue series. This is not impossible, since no residue lies ot the integraition contour and the zeroes of coshz are infinitely many, but countable.
I hope someone could help me with this puzzle
Any comments are appreciated.
Thanks a lot, Marine