= Parton on-shell delta identity
{title2=$2\delta((\xi P+q)^2)=\delta(\xi-x)/(P\cdot q)$}
In the leading <massless collinear parton approximation>, $P^2=0$ and $q^2=-Q^2$ imply $(\xi P+q)^2=2\xi P\cdot q-Q^2$. With <Bjorken x> $x=Q^2/(2P\cdot q)$ and $P\cdot q>0$, the <Dirac delta> change-of-variable formula yields the displayed identity. The positive-energy branch of the final <parton> also gives $\delta(E_q+\xi E_P-E_k)/E_k=2\delta((E_q+\xi E_P)^2-E_k^2)$ locally on physical support. Keeping target mass instead introduces $\xi^2M^2$ and changes the root and Jacobian.
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