  The Teukolsky equation has long been known to lead to divergent integrals
when it is used to calculate the gravitational radiation emitted when a test
mass falls into a black hole from infinity. Two methods have been used in the
past to remove those divergent integrals. In the first, integrations by parts
are carried out, and the infinite boundary terms are simply discarded. In the
second, the Teukolsky equation is transformed into another equation which does
not lead to divergent integrals. The purpose of this paper is to show that
there is nothing intrinsically wrong with the Teukolsky equation when dealing
with non-compact source terms, and that the divergent integrals result simply
from an incorrect choice of Green's function. In this paper, regularization of
the Teukolsky equation is carried out in an entirely natural way which does not
involve modifying the equation.
