Solid harmonics

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions formula_1. There are two kinds: the "regular solid harmonics" formula_2, which vanish at the origin and the "irregular solid harmonics" formula_3, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:
Derivation, relation to spherical harmonics.
Introducing "r", θ, and φ for the spherical polar coordinates of the 3-vector r, and assuming that formula_6 is a (smooth) function formula_1, we can write the Laplace equation in the following form
where "l"2 is the square of the nondimensional angular momentum operator,
It is known that spherical harmonics Yml are eigenfunctions of "l"2:
Substitution of Φ(r) = "F"("r") Yml into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution,
The particular solutions of the total Laplace equation are regular solid harmonics:
and irregular solid harmonics:
The regular solid harmonics correspond to harmonic homogeneous polynomials, i.e. homogeneous polynomials which are solutions to Laplace's equation.
Racah's normalization.
Racah's normalization (also known as Schmidt's semi-normalization) is applied to both functions 
(and analogously for the irregular solid harmonic) instead of normalization to unity. This is convenient because in many applications the Racah normalization factor appears unchanged throughout the derivations.
Addition theorems.
The translation of the regular solid harmonic gives a finite expansion,
where the Clebsch-Gordan coefficient is given by
The similar expansion for irregular solid harmonics gives an infinite series,
with formula_18. The quantity between pointed brackets is again a Clebsch-Gordan coefficient,
References.
The addition theorems were proved in different manners by several authors. For example, see the two different proofs in:
Real form.
By a simple linear combination of solid harmonics of ±"m" these functions are transformed into real functions, i.e. functions formula_20. The real regular solid harmonics, expressed in Cartesian coordinates, are real-valued homogeneous polynomials of order formula_21 in "x", "y", "z". The explicit form of these polynomials is of some importance. They appear, for example, in the form of spherical atomic orbitals and real multipole moments. The explicit cartesian expression of the real regular harmonics will now be derived.
Linear combination.
We write in agreement with the earlier definition 
with
where formula_24 is a Legendre polynomial of order "l".
The "m" dependent phase is known as the Condon-Shortley phase.
The following expression defines the real regular solid harmonics:
and for "m" = 0:
Since the transformation is by a unitary matrix the normalization of the real and the complex solid harmonics is the same.
"z"-dependent part.
Upon writing "u" = cos θ the "m"th derivative of the Legendre polynomial can be written as the following expansion in "u"
with
Since "z" = "r" cosθ it follows that this derivative, times an appropriate power of "r", is a simple polynomial in "z",
("x","y")-dependent part.
Consider next, recalling that "x" = "r" sinθcosφ and "y" = "r" sin θsin φ,
Likewise
Further
and
List of lowest functions.
We list explicitly the lowest functions up to and including "l = 5" .
Here formula_36
The lowest functions formula_38 and formula_39 are: