Dimensional Regularization in Peskin

[gobbles]

Homework Statement

I'm trying to understand dimensional regularization with Peskin. There is a transitions that is not clear.

Homework Equations

On page 250, the general expression for the d-dimensional integral is given:
$\int \frac{d^d l_E}{(2\pi)^d}\frac{1}{(l_E^2+\Delta)^2}=\frac{1}{(4\pi)^{d/2}}\frac{\Gamma(2-\frac{d}{2})}{\Gamma(2)}\left(\frac{1}{\Delta}\right)^{2-\frac{d}{2}}$.
So far everything is clear. But then, in 7.84 he writes
$\int \frac{d^d l_E}{(2\pi)^d}\frac{1}{(l_E^2+\Delta)^2}\rightarrow\frac{1}{(4\pi)^2}\left(\frac{2}{\epsilon}-\log\Delta-\gamma+\log(4\pi)+\mathcal{O}(\epsilon)\right),$
when $d\rightarrow4$.

The Attempt at a Solution

I understand where the $\frac{2}{\epsilon}$ and $-\gamma$ factors come from, but where did the terms involving the logarithm function came from? Even if I take the eventual integration over the Feynman parameters into account I don't get the correct answer.

 

[Orodruin]

Use the fact that $a^b = \exp(b \ln a)$ and expand the exponent in $\epsilon$. This gives a linear term proportional to the logs, which will be multiplied by the $1/\epsilon$ dependence from the gamma function and therefore give a term constant in $\epsilon$.

 

[gobbles]

Thank you! Haven't thought of going that way.