A general non-asymptotic solution is given for the quantum Yukawa scattering problem in terms of Legendre functions of the second kind. It is visualized and shown to match the standard result in the asymptotic limit. We also briefly indicate how the result contributes to three problems that have heretofore been obscured by the usual approximations: it clarifies the meaning of the differential cross section by providing a generalizing case, it illustrates the breakdown of the assumption that scattering currents are only radially outward which leads to fundamental interpretive issues, and it provides a way to incrementally and rigorously investigate the Coulomb limit of scattering.
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