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Homework Help: Integral using residue theorem

  1. Sep 23, 2011 #1
    The question asks to show using the residue theorem that

    [itex]\int[/itex]cos(x) / (x2 +1)2 dx = [itex]\pi[/itex] / e

    (the terminals of the integral are -∞ to ∞ but i didnt know the code to write that)

    I found the singularities at -i and +i

    so i think we change the function inside the integral to cos(z) / (z2 +1)2

    i expanded the cos(z) as cosh(1) - isinh(1)(z-i) -0.5cosh(1)(z-i)2 +...

    and i expanded (z2+1)2 as -(1/4)(z-i)2 - i/4(z-i) + 3/16 +...

    I did the same for the singularity at x=-i and when i added both the residues together i got

    (9/16e + e/16) (this is multiplied by 2[itex]\pi[/itex]i to find residues)

    this doesnt seem right? i dont know if what ive done is the right method. please help, ive spent soooo many hours on this one stupid question :(
  2. jcsd
  3. Sep 23, 2011 #2


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    why do you add singularities together? not too sure what you're attempting...

    i think the idea is you need to set up a closed contour in complex space and integrate around it. Then teh vlaue of teh integral will be related to singularities contained within the contour.

    a good one would be along the x axis for -R to R, and a semi-circle in the upper half of the complex plane to connect the contour. Then take the limit r->infinity. Hopefully the semi-cirlce contribution tends to zero, then you can relate the x axis component directly to the original integral
  4. Sep 23, 2011 #3


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    Wouldn't you have to do that first, in order to find the singularities?

    What path are you integrating over? Does it include both singularities? Obviously to do this real integral as a path integral, you need a closed path that includes the real axis. I don't see how you can do that and get both i and -i inside the path.
  5. Sep 23, 2011 #4
    oh yeah... that makes sense...

    your right we cant get i and -i in on a path integral across the real axis

    maybe i need to build a circle or something? i guess i have less of an idea now?

    thanks for your help?

    any idea where i should go from here?
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