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Integral with singularities at the endpoints

  1. Oct 29, 2009 #1
    1. The problem statement, all variables and given/known data

    Evaluate the Integral [tex]\int[/tex] dx/((a^2+x^2)*sqrt(1-x^2)) from -1 to 1
    Using contour integration

    2. Relevant equations

    Residue theorem/Cauchy integral forumula


    3. The attempt at a solution

    So I know that at the end-points of the interval (abs(z) = 1) that a singularity exists, so a branch-cut from -1 to 1 needs to be made. Additionally, the singularities in the contour used for the residue theorem are located at z = +ia and -ia. After this I am confused. I can't find any resources that describe how to evaluate a contour integral where the endpoints of the interval used to define the contour has branch-point singularities. Any ideas?

    I am thinking maybe the contour that I described ( from just above the real-axis counter-clockwise to real 1 and then up into the imaginary plane heading counter-clockwise towards -1 on the real axis enclosing the singularity z= +ia)

    Any thoughts?
     
  2. jcsd
  3. Oct 29, 2009 #2

    lanedance

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    Homework Helper

    haven't done complex analysis for a while, so take anything i say with a grain of salt...

    could you define the branch cut from [-1 , 1] on the real axis then all then integrate all the way around the branch cut, with little loops that shrink to zero around the branch points.

    The argument of the function will change as you go round the loop, which should help you evaluate the integral. And the value of the loops should go to zero because of the power of the singularity

    if that doesn't work, you could try the branch cuts (-infinty,-1] & [1,infinty) and maybe try a semi circle and a full circle (sort of, but with little loop round the cuts)
     
  4. Oct 30, 2009 #3
    Ok, I've got some new info to add, but I am not sure how to use it.

    So I figured out that the contour should have the form of a dogbone/dumbbell encircling the branch-cut from -1 to 1 counterclockwise. But I have no idea how to deal with this. I've gone over branch-cuts only once in class, so working with this problem has kind of been a pain.

    Any help would be great
     
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