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Contour integral w/ sqrt

  1. Oct 10, 2011 #1
    1. The problem statement, all variables and given/known data
    The integral I'm trying to solve is sqrt(x)/(1+x^2) from 0 to infinity.


    2. Relevant equations



    3. The attempt at a solution I've attached my solution. I know it's not right, as I shouldn't get an imaginary solution. The answer is actually pi/sqrt(2) according to my book. The author used a keyhole contour (branch cut along positive real axis and avoiding the branch point at z=0), but I don't see what's wrong with my approach. Any help would be greatly appreciated.
     

    Attached Files:

  2. jcsd
  3. Oct 10, 2011 #2

    Mute

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    Homework Helper

    First consider: What is [itex]i^{1/2}[/itex] in cartesian form? (x + i*y). This won't solve the issue, but is just to remind you that to check that i^0.5 isn't actually 1- i (it's not, but it's close).

    The real problem is the arc on the negative axis. You set [itex]z = re^{i\pi}[/itex], but forgot an extra factor of e^{i pi} from the differential dz.
     
    Last edited: Oct 10, 2011
  4. Oct 10, 2011 #3
    sqrt(i) = e^(i*pi/4)= (1/sqrt(2))*(1+i)... Any idea what is amiss? I still don't see it.
     
  5. Oct 10, 2011 #4

    Mute

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    Whoops, you got in before I edited it. You missed a factor of e^{i pi} that comes from [itex]dz = dr e^{i\pi}[/itex] on your arc on the negative axis. You really have a factor of [itex](1-e^{i 3 \pi/2}) = (1 + i)[/itex].
     
  6. Oct 10, 2011 #5
    Ah, okay. Thanks, so much. That solves some of the issues I was having with earlier problems too I think.
     
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