Residue theorem for real integrals

[shebbbbo]

The question asks to show using the residue theorem that

\int cos(x)/(x²+1)² dx = \pi/e
(the terminals of the integral are -\infty to \infty but i didnt know the code to write that)

I found the singularities at -i and +i

so i think we can then say

\intcos (z) / (z+i)²(z-i)² dz

and if we take that integral over a closed contour then we will be left with the residue

and i know that once we have found the residue you can multiply this by 2\pii and sum all the residues together.

BUT...

i don't know how to find the residues for this question. the cos on the top line is causing me trouble. maybe i need to expand the cos as a series? but I am not sure

thanks

 

[Hootenanny]

There are two ways to compute the residue of poles - series or via limits and derivatives. See here: http://en.wikipedia.org/wiki/Residue_(complex_analysis)#Calculating_residues

 

[shebbbbo]

ive looked on that page and stared at it for ages...

will both methods work? and am i correct with what i have done so far? (ie poles at +i and -i)?

 

[Hootenanny]

ive looked on that page and stared at it for ages...

will both methods work? and am i correct with what i have done so far? (ie poles at +i and -i)?

Yes, both methods will work and you are indeed correct in what you have done so far.