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 . The method also provides a direct route to the capacitance of a slender torus, without the use of toroidal coordinates.