One-electron Green s functions play a central role in multiple-scattering theory (MST) based electronic- structure methods. Robust methods exist for calculating the Green s function for crystal potentials that are spherically symmetric about atomic centers. When applied to potentials of general shape, these same techniques result in pathologies in the small-r behavior of the electronic charge density because a portion of the Green s function can become singular at the origin for that case. We propose an algebraic method that eliminates the singular behavior by making use of the equivalence of two terms that involve poles in the inverse of the sine matrix. Our accurate calculations illustrate the limitations of previous methods for treating this problem that rely on extrapolating the solutions near the origin.