# Monotonic properties of the shift and penetration factors. (arXiv:1708.07554v2 [quant-ph] UPDATED)

with respect to energy, angular momentum, and charge. Definitive results for

the signs of these derivatives are found for the repulsive Coulomb case. In

particular, we find that the derivative of the shift factor with respect to

energy is positive for the repulsive Coulomb case, a long anticipated but

heretofore unproven result. These results are closely connected to the

properties of the sum of squares of the regular and irregular Coulomb

functions; we also present investigations of this quantity.