The modified Cheng representation (MCR) is evaluated for $S$, $P$, and $D$ waves with the potentials $\ensuremath{-}\frac{1.8{e}^{\ensuremath{-}r}}{r}$ and $\ensuremath{-}\frac{5{e}^{\ensuremath{-}r}}{r}$, the former being strong enough to have one bound state near the $S$-wave threshold, while the latter is nearly strong enough to have a $P$-wave resonance. We find good agreement with exact results with only one trajectory input, the agreement being better than the Cheng representation (CR) with three trajectories. Although both representations, the MCR and the CR, are automatically unitary, the results are a considerable improvement over the earlier modified Khuri representation, which was not unitary---especially for the more attractive potential $\ensuremath{-}\frac{5{e}^{\ensuremath{-}r}}{r}$. Residues above threshold are also calculated and compared with the exact results, while reduced residues both above and below threshold are calculated via the MCR, and in all cases good agreement with exact results is obtained. Finally, a simple approximation to the reduced residue below threshold is noted to be quite accurate in potential theory, and may provide a simple and accurate means of estimating high-energy total cross sections and diffraction widths.