Vanishing angular singularity limit to the hard-sphere Boltzmann equation

In this note we study Boltzmann's collision kernel for inverse power law interactions $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $d=3$. We prove the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near $\theta\simeq 0$ in the limit $s\to \infty$. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.