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∫dx/((x^(2/3)(x+1)), integrated over [0,∞]

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data

    As in thread title.

    2. Relevant equations

    Residue Theorem.

    3. The attempt at a solution

    I just need help figuring out the circle C I'll be using. Suggestions?
     
  2. jcsd
  3. Feb 6, 2012 #2

    vela

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    What does the presence of z2/3 tell you?
     
  4. Feb 6, 2012 #3
    Other than that there's a pole at z=0?
     
  5. Feb 6, 2012 #4

    vela

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    Yes, other than that. In particular, what's the effect of the fractional power?
     
  6. Feb 6, 2012 #5
    Change the distance between z and the origin from r to r2/3
    Change the angle between z and the x-axis from ø to 2ø/3
     
  7. Feb 6, 2012 #6

    vela

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    Right. Do you know what a branch point and a branch cut are?
     
  8. Feb 6, 2012 #7
    Yeah, I somehow need a loop that avoid z=-1 and z=0. Right?
     
  9. Feb 6, 2012 #8

    vela

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    It's more that you want to avoid crossing the branch cut than avoiding z=0, and you obviously want a piece or pieces of the contour to correspond to the original integral.
     
  10. Feb 6, 2012 #9
    So I'd take R>1 and make a half circle of radius R in the upper half of the plane. Then I'd make two little half circles that jump over z=-1 and z=0. Then I'd look at ∫C f(z)dz as the sum of several integrals, one of which can written as a real-valued integral and see what happens as R→∞ and the radii of the little half circles go to zero. Right?
     
  11. Feb 6, 2012 #10

    vela

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  12. Feb 6, 2012 #11
  13. Feb 6, 2012 #12

    vela

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    Doesn't the answer to that question depend on which way Pacman is moving?
     
  14. Feb 6, 2012 #13
    I forgot that PacMan is in perpetual motion.

    But yeah, how am I gonna do this? I need C to be formed from a series of paths, each of which will have a line integral that approaches a real value after I take some limit.
     
  15. Feb 6, 2012 #14

    vela

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    Start by taking the keyhole contour and break it into four pieces and evaluate the line integral for each piece.
     
  16. Feb 6, 2012 #15
    How would that work? I want ∫f(x)dx (integrated on [0, R]) to be one of the four line integrals.
     
  17. Feb 8, 2012 #16

    vela

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    That's what you're supposed to figure out. :smile: Did you understand the example on Wikipedia? That's pretty much the recipe you want to follow.
     
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