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Contour integration problem

  1. Nov 11, 2009 #1
    Hello! I wanted to solve this integral but really didn't understand the method show in the book.
    Can anyone help me out please.

    sorry I don't know how to show the integral sign, here it is

    integral of - to + infinity (Cos(x)/x^2 + 2x +5 )dx

    here Cos(z) = Re[exp(iz)]

    I tried to reduce the denominator to a (a+b)^2 thing and then use a method that I didn't understand ( so I copied it from the book).
    I need to know the stepwise solution if possible.

    Thank You!
     
  2. jcsd
  3. Nov 11, 2009 #2

    HallsofIvy

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    I presume you mean you wrote [itex]x^2+ 2x+ 5= x^2+ 2x+ 1+ 4= (x+1)^2+ 4[/itex]. The function [itex]e^z/((z+1)^2+ 4[/itex] is analytic everywhere except at z= -1+2i and -1-2i where there are simple poles.

    If you take your contour to be along the x-axis from (-R, 0) to (R, 0), then around the half circle in the upper half plane from (R, 0) to (-R, 0), the integral is the residue at -1+ 2i divided by [itex]2\pi i[/itex]. The integral you want is the integral on the x axis, as R goes to infinity. If you can find the integral around [itex]z= Re^{i\theta}[/itex] as [itex]\theta[/itex] goes from 0 to [itex]\pi[/itex], you can subtract that from the inegral around the contour, for any R, to get the integral along the x-axis..
     
  4. Nov 11, 2009 #3
    thank you very much.
     
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