In physics, spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance to a source, i.e., as Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential. For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density Through this article, the primed coordinates such as refer to the position of charge(s), whereas the unprimed coordinates such as refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector has coordinates where is the radius, is the colatitude and is the azimuthal angle. Spherical multipole moments of a point charge The electric potential due to a point charge located at is given by where is the distance between the charge position and the observation point and is the angle between the vectors and . If the radius of the observation point is greater than the radius of the charge, we may factor out 1/r and expand the square root in powers of using Legendre polynomials This is exactly analogous to the axial multipole expansion. We may express in terms of the coordinates of the observation point and charge position using the spherical law of cosines (Fig. 2) Substituting this equation for into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem where the functions are the spherical harmonics. Substitution of this formula into the potential yields which can be written as where the multipole moments are defined As with axial multipole moments, we may also consider the case when the radius of the observation point is less than the radius of the charge. In that case, we may write which can be written as where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics The two cases can be subsumed in a single expression if and are defined to be the lesser and greater, respectively, of the two radii and ; the potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion Exterior spherical multipole moments It is straightforward to generalize these formulae by replacing the point charge with an infinitesimal charge element and integrating. The functional form of the expansion is the same. In the