  We study derivatives of the shift and penetration factors of collision theory
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.
