  Gravitational perturbations due to a point particle moving on a static black
hole background are naturally described in Regge-Wheeler gauge. The first-order
field equations reduce to a single master wave equation for each radiative
mode. The master function satisfying this wave equation is a linear combination
of the metric perturbation amplitudes with a source term arising from the
stress-energy tensor of the point particle. The original master functions were
found by Regge and Wheeler (odd parity) and Zerilli (even parity). Subsequent
work by Moncrief and then Cunningham, Price and Moncrief introduced new master
variables which allow time domain reconstruction of the metric perturbation
amplitudes. Here I explore the relationship between these different functions
and develop a general procedure for deriving new higher-order master functions
from ones already known. The benefit of higher-order functions is that their
source terms always converge faster at large distance than their lower-order
counterparts. This makes for a dramatic improvement in both the speed and
accuracy of frequency domain codes when analyzing unbound motion.
