Method of residues integration problem

[thrillhouse86]

Hi all,

I am trying to determine two Melnikov Integrals (after some manipulation) of the form:
\int^{\infty}_{-\infty}cos(at)sech(bt) dt
and
\int^{\infty}_{-\infty}cos(at)sech^{3}(bt) dt

The textbook I've been reading (Litchenberg & Libermann), says that the way to integrate similar problems is to use the method of residues. I have a superficial understanding of how that works, but every other time I've used the method of residue I've dealt with rationale functions so that I could find poles and then just apply apply the Cauchy integral formula at the isolated poles.

Can anyone give me a kick-start on how to begin solving these 2 integrals by the method of residues ?

[Marin]

Hi thrillhouse86,

I'm not an expert on the "residue Issue" but here's my suggestion:

1. Note that both integrands are even functions (at least I think so)
2. Then complexify the cos-functions, i.e. use the e-function

3. For the Residue Thm. you will need a special contour. I had posted a similar thread a month ago, I'd try with the same contour:

http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/99385-contour-integral-residue-thm.html


PS: post the answer when you have it, I'm interested too :)