evolution equation $$\frac{\partial F(x,Q^2)}{\partial \textrm{ln}Q^2} = \frac{4 \alpha _s}{3 \pi} \int
dzdx'\hat{P}(z)F(x',Q^2)\delta (x-zx')$$ Here we have replaced $P$ with $\hat{P}$ (we describe $\hat{P}$ below), and using $\hat{P}$ one can let $\epsilon \rightarrow 0$. This can be done since the divergence can be treated by defining a distribution that can be integrated by subtracting a delta function from the singular term. We define the distribution, $\frac{1}{(1-z)_+}$, such that for any given smooth function $f(z)$ we have $$\int_{0}^{1} dz\frac{f(z)}{(1-z)_+} = \int_{0}^{1} dz\frac{f(z) - f(1)}{(1-z)}$$ and then we replace the splitting function $P$ with $\hat{P}$ given by $$\hat{P} = \frac{1}{2} \frac{1+z^2}{(1-z)_+} + \frac{3}{2} \delta (1-z)$$

In fact, it is $\hat{P}$ and not $P$ that appears in the AP equations. The delta function comes from the fact that we have to take into account that a parent quark splitting into a quark and a gluon disappears. Thus, we get contributions from the new particles but the one in the initial state no longer exists, and this must be taken into account. The coefficient in front of the delta function, $3/2$, is found by the requirement that the number of quarks is conserved, which means that the integral of $\hat{P}$ over $z$ must be zero. 