I Loop Integral Form: Finding a Workable Solution without Regularization

[DeathbyGreen]

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)

<br /> \int_x^{\infty}du \frac{u^2}{\omega - u}<br />

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!

 

[NFuller]

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 $u=w$.

 

[DeathbyGreen]

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?

 

[NFuller]

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.

 

[aheight]

The integral converges in the Cauchy Principal Valued sense. For example:

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