A new method for directly sampling the neutron resonance upscattering effect is presented. Alternatives have relied on inefficient rejection sampling techniques or large tabular storage of relative velocities. None of these approaches, which require pointwise energy data, are particularly well suited to the windowed multipole cross section representation. The new method called multipole analytic resonance scattering (MARS) overcomes these limitations by inverse transform sampling from the target relative velocity distribution where the cross section is expressed in the multipole formalism. The closed form relative speed distribution contains a novel special function we deem the incomplete Faddeeva function: $$ w(z, x) = \frac{i}π \int_{-\infty}^x \frac{e^{-t^2} dt}{z-t}. $$ We present the first results on its efficient numerical evaluation.