Radial wavefunction ($R_{nl}(r)$)

Separation of variables in a central potential writes a wavefunction as $R_{nl}(r)Y_l^m(\theta ,\varphi )$. With normalized spherical harmonics the radial factor satisfies ${\int }_0^{\infty }{r}^2|R_{nl}{|}^{2}dr=1$. The reduced radial function is $u=rR$, with normalization $\int |u{|}^{2}dr=1$; the two functions have different behavior at the origin. For a regular Coulomb eigenstate, $R_{nl}\propto r^l$ near the origin.