Hi there!

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

Observe that

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