The evolution of quark parton distribution functions (pdf) can be described as a resummation of gluon emissions from scale $Q_0^2 \rightarrow Q^2$ via DGLAP equation (in valence sector):

$$ \mu^2 \frac{\text{df}(x,\mu^2)}{\text{d}\mu^2} = \int_x^1\frac{dy}{y}P_{qq}\big(\frac{x}{y}, \alpha_s\big)\text{f}(y, \mu) $$

where to leading order $P_{qq}(z) = \frac{\alpha_sC_f}{2\pi} \big[\frac{1+z^2}{(1-z)_+} + \frac{3}{2}\delta(1-x)\big] + \mathcal{O}(\alpha_s^2)$.

To simplify things, we can consider moments of the pdf and study their evolution. Defining $a_{n+1}(\mu^2)=\int_{-1}^{1} dx\;x^n\;\text{f}(x,\mu^2)$ we find

$$ \mu^2 \frac{\text{d}a_{n+1}(\mu^2)}{\text{d}\mu^2} = \gamma_{qq}^{n+1}(\alpha_s)\;a_{n+1}(\mu^2) $$ where $\gamma_{qq}^{n} = \frac{\alpha_sC_f}{2\pi} \big[\frac{3}{2} - \frac{2}{n(n+1)} - 2\sum_{k=1}^n\frac{1}{k}\big]$.

Beyond this is where I need some help. Something I would like to do is to actually compute the inverse Mellin Transform

$$ \mu^2 \frac{\text{df}(x,\mu^2)}{\text{d}\mu^2} = \frac{1}{2\pi i}\int_{x-i\infty}^{x+i\infty}dn \; x^{-n}\gamma_{qq}^{n}(\alpha_s)\;a_{n}(\mu^2) $$

and recover the original DGLAP evolution equations. I have no idea how to do this integral. Can anyone help?


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