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Contour integration with a branch cut

  1. Feb 28, 2017 #1
    1. The problem statement, all variables and given/known data
    -11 dx/(√(1-x2)(a+bx)) a>b>0

    2. Relevant equations
    f(z0)=(1/2πi)∫f(z)dz/(z-z0)

    3. The attempt at a solution
    I have absolutely no idea what I'm doing. I'm taking Mathematical Methods, and this chapter is making absolutely no sense to me. I understand enough to tell I'm supposed to do contour integration on this with a branch cut on the singularity, but actually doing it is another thing. Also, I have no idea what to do with the second term in the denominator. If you can explain this to me, I would be grateful; and please, try to dumb it down. I can't even figure out how to find residues.

    The farthest I got was K=∫-11dx/(√(1-x2)(a+bx)) + lim r->∞0π reidθ/(√(1-r2ei2θ)(a+bre))
    I stopped there, however, because I'm fairly certain I'm embarking on several hours of barking up the wrong tree.
     
  2. jcsd
  3. Feb 28, 2017 #2

    strangerep

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    Have you had a proper course, or part-course, on contour integration? Without that, this will be very difficult.

    I could tell you that you're supposed to place a branch cut between ##x = \pm 1##, and use a "dog bone contour" (aka "dumbbell contour"), but that won't be much help if you don't know how to do easier contour integrals and compute basic residues. [Google for "dog bone contour" to see what this looks like.]
     
  4. Mar 1, 2017 #3

    vela

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    The OP is taking a math methods course right now and learning about contour integration and complex analysis right now.
     
  5. Mar 1, 2017 #4

    vela

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    As strangerep suggested, you should probably go back to trying to do easier problems first and getting a handle on those before trying to tackle this one.

    You could try finding a similar but simpler example in your textbook (one singularity, single branch cut) and asking questions about that first.
     
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