{If} one replaces the Poisson kernel of the unit disc by its square root, then normalised Poisson integrals of $L^{p}$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning ($1\leq p<\infty$) and Sjögren ($p=1$ and $p=\infty$). In this paper we present new and simplified proofs of these results. We also generalise the $L^{\infty}$ result to higher dimensions.
