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Contour integration problem-(sinx/x)^2

  1. Oct 1, 2011 #1
    contour integration problem--(sinx/x)^2

    1. The problem statement, all variables and given/known data
    I am to evaluate the integral of (sinx/x)^2 from -infinity to +infinity.


    2. Relevant equations



    3. The attempt at a solution

    I drew my contour as a large half circle in UHP, and this contour then includes the singularity at the origin.

    Using a trig identity, I can rewrite sin^x : 0.5(1-cos(2x))

    Since we're working in the UHP, I then rewrite the integral: Real part of 0.5*int((1-e^(2iz))/z^2)dz from -inf to +inf

    There is a second order pole at the origin. The residue here is a-= 0.5*(d/dz){(z^2*(1-e^(2iz)))/z^2)} evaluated as z-->0, = 0.5*-2i*e^(2i*0)=-i.

    Since the part enclosing the singularity is a half circle, the integral should be pi*i*a-=pi. But the answer is supposed to be pi/2.

    I apologize for the lack of LaTeX. Could somebody please give me a hint as to what I'm doing wrong? Thanks.
     
  2. jcsd
  3. Oct 1, 2011 #2
    Re: contour integration problem--(sinx/x)^2

    I now see that the error could possibly be that in converting the cosine to exponential form, I missed a factor of 1/2. But how come I have solved other problems correctly without that factor? See for example, the problem right before "Solution 13.18" on this document: http://www.cacr.caltech.edu/~sean/applied_math.pdf [Broken]
     
    Last edited by a moderator: May 5, 2017
  4. Oct 1, 2011 #3
    Re: contour integration problem--(sinx/x)^2

    That's just a simple pole. Expand it as a power series to see that.
     
  5. Oct 1, 2011 #4
    Re: contour integration problem--(sinx/x)^2

    Ah, okay, indeed it is a simple pole. But when I do the problem again, I still get pi. See the attached pdf.
     
  6. Oct 1, 2011 #5
    Re: contour integration problem--(sinx/x)^2

    Here's the pdf:
     

    Attached Files:

  7. Oct 1, 2011 #6
    Re: contour integration problem--(sinx/x)^2

    I get pi too and that's what Mathematica gives also.
     
  8. Oct 1, 2011 #7
    Re: contour integration problem--(sinx/x)^2

    Hmmm. Arfken & Weber problem 7.1.12 asks me to prove that it's pi/2. Possible typo?
     
  9. Oct 1, 2011 #8

    vela

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    Re: contour integration problem--(sinx/x)^2

    In my copy of Arfken (3rd edition), it's problem 7.2.12, and the limits on the integral are from 0 to infinity.
     
  10. Oct 1, 2011 #9
    Re: contour integration problem--(sinx/x)^2

    It must be a typo then. In the 6th edition, the limits are from -infinity to +infinity.
     
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