Series reduction method for scattering of planar waves by Kerr black holes

We present a practical method for evaluating the scattering amplitude f8(θ,ϕ) 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 f8 is a divergent series, but f8 itself diverges only at a single point on the sphere. Here we show that f8 can be expressed as the product of a reduced series and a prefactor 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.