Convergence of the bipolar expansion for the coulomb potential



The bipolar expansion of the Coulomb potential, which underlies the multipole moment expansion for interacting charge distributions, converges like a geometric series for separated charges, but converges at best only conditionally when the charges interpenetrate. This article shows how the order of summation affects the sum. Evidence is presented for simpler series when the geometry is linear. © 2014 Wiley Periodicals, Inc.