Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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##
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Loading...
Similar Threads for Loop Integral Form Date
I How to calculate the area under a curve Jul 21, 2017
Closed loop contour Nov 4, 2014
Weird ways of doing closed loop integrals Jul 21, 2013
Area of loops and arcs of prolate cycloids Jul 12, 2013
Loop Integral May 27, 2010