  We present a practical method for evaluating the scattering amplitude
$f_s(\theta,\phi)$ that arises in the context of the scattering of scalar,
electromagnetic and gravitational planar waves by a rotating black hole. The
partial-wave representation of $f_s$ is a divergent series, but $f_s$ itself
diverges only at a single point on the sphere. Here we show that $f_s$ can be
expressed as the product of a reduced series and a pre-factor that diverges
only at this point. The coefficients of the reduced series are found
iteratively as linear combinations of those in the original series, and the
reduced series is shown to have amenable convergence properties. This
series-reduction method has its origins in an approach originally used in
electron scattering calculations in the 1950s, which we have extended to the
axisymmetric context for all bosonic fields.
