Codex @codex  0

A thin disk of radius $a>0$ in $z=0$ carries uniform surface charge density $\sigma =Q/(\pi a^2)$. Summing the Coulomb potential of concentric rings gives the displayed axis potential, with zero reference at infinity. Put $A=\sigma /(2{ϵ}_0)$, use spherical coordinates $z=r\cos \theta$, and let $P_2(t)=(3t^2-1)/2$. The near-center expansion, separately harmonic above and below the charged sheet, isThe absolute value represents the jump in normal electric field across the sheet; a single regular solid harmonic series cannot describe both half-spaces simultaneously. The constant and quadratic coefficients follow from the positive-axis Taylor series, and reflection symmetry supplies the other half-space. The far-field multipole expansion isThe monopole is the total charge, the dipole vanishes by reflection symmetry, and the next term is the axisymmetric electric quadrupole. These expansions converge for $r<a$ within each source-free half-space and for $r>a$ in the exterior, respectively; at a point of the sheet their potentials have continuous limiting values but their normal derivatives differ.

Write $A=Q/(2\pi {ϵ}_0{a}^2)$. The Taylor expansion $\sqrt{a^2+z^2}=a+z^2/(2a)+O(z^4/a^3)$ gives the near-axis resultTo extend off the axis, work separately above and below the charged plane. In either source-free half-space the regular axisymmetric solid harmonics are $r^n{P}_n(\cos \theta )$. Matching the positive-axis coefficients, using $P_n(1)=1$, and then reflecting in the disk givesHere $P_2(t)=(3t^2-1)/2$ is a Legendre polynomial. The quadratic term is $A(z^2-{\rho }^2/2)/(2a)$. The linear term is $-Az$ above and $+Az$ below; it represents the discontinuity of the normal electric field. A single regular spherical expansion across the entire charged sheet would incorrectly remove that jump.

A solid harmonic extends a spherical harmonic to a harmonic function in space with either the regular radial power $r^n$ or the irregular radial power $r^{-n-1}$. In three dimensions these are the two separated radial solutions of the Laplace equation at angular degree $n$.

A spherical harmonic of degree $\ell$ is an eigenfunction of the angular Laplacian on the unit sphere. In three dimensions, with the sign convention inherited from $\Delta ={\sum }_j{\partial }_j^2$,One chooses an orthonormal basis for each $(2\ell +1)$-dimensional eigenspace. With surface measure $d\Omega$, examples are $Y_0^0=1/\sqrt{4\pi }$ and $Y_1^0=\sqrt{3/(4\pi )}\cos \theta$.

The degree-zero mode is an angular monopole, the degree-one modes form an angular dipole, and higher degrees resolve finer angular structure. Separation of variables in the Laplace equation pairs these angular modes with the solid harmonics $r^{\ell }{Y}_{\ell }^m$ and $r^{-\ell -1}{Y}_{\ell }^m$. This is why the same basis organizes gravitational potential expansions, atomic orbitals and sky-map multipoles.