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I Loop Integral Form

  1. Jan 3, 2018 #1
    Hi,

    I'm trying to calculate an integral which looks unfortunately divergent. The structure is similar to a loop integral but the appendix in the Peskin textbook didn't have a useable form. The integral form is (I did a u substitution to make it easier to look at)

    [itex]
    \int_x^{\infty}du \frac{u^2}{\omega - u}
    [/itex]

    Does anyone know of a workable form for this? Introducing a cutoff is possible but I would prefer not to.

    Edit: I know that it is divergent. I was hoping for some sort of regularization technique which would allow for a finite answer under certain conditions.
    Thank you!
     
    Last edited: Jan 3, 2018
  2. jcsd
  3. Jan 3, 2018 #2
    Yes, the integral is divergent as stated. What is the context of this problem? The title says this is a loop integral. If this is indeed a contour integral, then it can be evaulated on the complex plane about the pole at ##u=w##.
     
  4. Jan 3, 2018 #3
    The integral occurs in a conductivity calculation I'm working through, it has a imaginary convergence factor in the denominator which I didn't write. So you think I could evaluate this as a standard contour and not need the fancy QFT loop integral forms?
     
  5. Jan 3, 2018 #4
    It depends on what you are trying to evaluate. Without seeing the problem, I am not sure if the integral has been constructed correctly. If your goal is to evaluate a closed line integral, then the integral is zero if the loop does not enclose the pole and is ##2\pi i\text{Res}(f)## if it encloses the pole.
     
  6. Jan 6, 2018 #5
    The integral converges in the Cauchy Principal Valued sense. For example:

    ##\text{P.V.}\displaystyle\int_0^2 \frac{z^2}{1-z}dz=-4##
     
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