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Integrating with indented contour

  1. Dec 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Evaluate the following integral by integrating the corresponding complex function.

    [itex]\int_{-\infty}^\infty \frac{dx}{x(x^2+x+1)}[/itex]

    2. Relevant equations

    Cauchy's Residue Theorem for simple pole at a:[itex]Res(f;a)=\displaystyle\lim_{z\rightarrow a} (z-a)f(z)[/itex]

    3. The attempt at a solution
    I have used the definite real integral widget on wolfram which states that the integral does not converge. Will I be able to show this is the case by integrating around the semi circular contour indented at 0?
  2. jcsd
  3. Dec 9, 2013 #2
    Did you look at it first? I mean plot it say from -10 to 10? Looks to me it has the right shape to converge in the Cauchy Principal Value sense. That is the only way it can converge since it has a pole on the path of integration. Alpha is telling you it diverges in the Riemann sense. Did you try:

    Code (Text):

    Integrate[1/(x*(x^2 + x + 1)),
      {x, -Infinity, Infinity}, PrincipalValue -> True]
    However, if you're not familiar with Principal-valued integrals, you may want to look that up.

    Edit: made a mistake with the function. It's +1 and I corrected it above but still everything I said in regards to the function with -1 applies to this function as well.
    Last edited: Dec 9, 2013
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