# Integral Identity

Hey folks!

I'm trying to figure out an identity from a paper on dimensional regularization.

Here's the identity:

$$-\frac{1}{2}\frac{d}{ds}|_{s=0}\int_0^\infty \frac{d^4k}{(2\pi)^4}(k^2+m^2)^{-s}$$

after performing the k-integral this becomes:

$$=-\frac{1}{32\pi^2}\frac{d}{ds}|_{s=-2}\frac{1}{s(s+1)}m^{-2s}$$

I found this in a paper with no references. Is this perhaps something out of Gradstein and Ryzhik?

Last edited:

Mute
Homework Helper
There's a problem with the expression you've written. s appears on the left hand side, but on the right hand side it looks like after taking the derivative you're setting all the s's to 2. If s appears as a variable/parameter on the left hand side it has to appear on the right hand side, so presumably at least one of the s's on the right hand side is the same as on the left hand side. (or the s on the left hand side should be a 2). If you could clarify this it would be of some help.

Hi Mute - I think I was modifying the expression while you were kindly looking at it. This one's good.

Mark44
Mentor
By "k-integral" do you mean this:
$$-\frac{1}{2}\frac{d}{ds}|_{s=0}\int_0^\infty \frac{d^4k}{(2\pi)^4}(k^2+m^2)^{-s} dk$$
?
I suppose this makes more sense:
$$-\frac{1}{2}\frac{d}{ds}|_{s=0}\int_0^\infty \frac{d^4k}{(2\pi)^4}(k^2+m^2)^{-s} ds$$

...anyone?