1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Integrating with indented contour
  1. Contour integral (Replies: 1)

  2. Contour integration (Replies: 4)

  3. Contour integrals (Replies: 2)

Loading...