State the convention before computing a multipole

On the unit sphere, spherical harmonics are eigenfunctions of the angular Laplacian. Degree l is a nonnegative integer and order m runs from −l to +l. We use unit integral norm, the usual Condon–Shortley phase, and the conjugation relation between opposite m values. A real potential may use complex coefficients, provided they obey the corresponding conjugation condition. Complex notation does not turn the observable electrostatic potential into a complex physical quantity.

Separation of the charge-free spherical Laplace equation gives radial powers rˡ and r⁻ˡ⁻¹ for each harmonic. Regular interior solutions discard the inverse powers at the origin. Exterior solutions that decay at infinity discard growing powers. Neither choice applies automatically in a shell between two finite radii, where both branches may be needed. The domain and boundary conditions decide the allowed radial functions before coefficients are fitted.

For charges confined inside radius a, define the spherical moment by integrating charge density times rˡ and the conjugated harmonic. The exterior potential then has the displayed coefficient divided by permittivity and by 2l+1. This formula includes the factor of four pi through the normalization of the harmonics. Replacing them with unnormalized Legendre polynomials requires changing coefficients as well; one cannot swap angular functions while retaining the same numerical moments.

For l=0 the coefficient is total charge divided by the square root of four pi, recovering the ordinary Coulomb potential. For an axisymmetric dipole, the l=1,m=0 coefficient is the z-directed dipole moment times the square root of three over four pi. The leading nonzero multipole dominates far away, but dominance is an approximation unless all higher moments vanish. Changing the expansion origin changes higher moments; even the dipole depends on origin when net charge is nonzero.

If potential is prescribed on a sphere rather than charge density inside it, project the surface voltage onto the harmonics. Each surface coefficient then multiplies the appropriate exterior power of a/r. Those surface coefficients have volts as units; they are not the source moments, whose units depend on l. A boundary with only constant and cosine angular terms produces an exact monopole-plus-dipole exterior potential under the stated decay condition.

The spectral lab illustrates coefficient bookkeeping in a simpler basis. Reset and set N=2 and p=0; predict equal amplitudes of its two sine terms. Reset and set N=2 and p=2; predict that the second coefficient becomes one quarter of the first. Truncation changes angular or spatial detail in many expansions, but this instrument uses one-dimensional sin(nx−t), not spherical harmonics, multipole moments, or the radial powers derived here.