Axially symmetric charge distributions and the arithmetic-geometric mean

The potential at an arbitrary point in space due to an axially symmetric charge distribution is related to the arithmetic-geometric mean of the maximum and minimum distances from each annulus of constant charge density. The arithmetic-geometric mean is expressible in terms of the elliptic integral of the first kind, K. Thus the potential of a charged body with cylindrical symmetry is reducible to a double integral over the charge density times K. For conductors the charge resides on the surface, and the potential reduces to a single integral over the surface charge density times K. This result leads to a new proof of the relation between a sum over products of Legendre polynomials and the complete elliptic integral of the first kind, and to new identities for the angular average of Legendre polynomials divided by |r−r′|. The method also provides a direct route to the capacitance of a slender torus, without the use of toroidal coordinates.