Spherical harmonics

Pierre-Simon Laplace, 1749–1827

Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his Mécanique Céleste, determined that the gravitational potential ${\displaystyle \mathbb {R} ^{3}\to \mathbb {R} }$ at a point x associated with a set of point masses m_i located at points x_i was given by

$${\displaystyle V(\mathbf {x} )=\sum _{i}{\frac {m_{i}}{|\mathbf {x} _{i}-\mathbf {x} |}}.}$$

Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r₁ = |x₁|. He discovered that if r ≤ r₁ then

$${\displaystyle {\frac {1}{|\mathbf {x} _{1}-\mathbf {x} |}}=P_{0}(\cos \gamma ){\frac {1}{r_{1}}}+P_{1}(\cos \gamma ){\frac {r}{r_{1}^{2}}}+P_{2}(\cos \gamma ){\frac {r^{2}}{r_{1}^{3}}}+\cdots }$$

where γ is the angle between the vectors x and x₁. The functions ${\displaystyle P_{i}:[-1,1]\to \mathbb {R} }$ are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle γ between x₁ and x. (See Applications of Legendre polynomials in physics for a more detailed analysis.)

In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. The solid harmonics were homogeneous polynomial solutions ${\displaystyle \mathbb {R} ^{3}\to \mathbb {R} }$ of Laplace's equation

$${\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.}$$

By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. (See the section below, "Harmonic polynomial representation".) The term "Laplace's coefficients" was employed by William Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had properly been introduced by Laplace and Legendre.

The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.

The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The (complex-valued) spherical harmonics ${\displaystyle S^{2}\to \mathbb {C} }$ are eigenfunctions of the square of the orbital angular momentum operator

$${\displaystyle -i\hbar \mathbf {r} \times \nabla ,}$$

and therefore they represent the different quantized configurations of atomic orbitals.

Real (Laplace) spherical harmonics ${\displaystyle Y_{\ell m}}$ for ${\displaystyle \ell =0,\dots ,4}$ (top to bottom) and ${\displaystyle m=0,\dots ,\ell }$ (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics ${\displaystyle Y_{\ell (-m)}}$ would be shown rotated about the z axis by ${\displaystyle 90^{\circ }/m}$ with respect to the positive order ones.)

Alternative picture for the real spherical harmonics ${\displaystyle Y_{\ell m}}$.

Laplace's equation imposes that the Laplacian of a scalar field f is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function ${\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} }$.) In spherical coordinates this is:[2]

$${\displaystyle \nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.}$$

Consider the problem of finding solutions of the form f(r, θ, φ) = R(r) Y(θ, φ). By separation of variables, two differential equations result by imposing Laplace's equation:

$${\displaystyle {\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=\lambda ,\qquad {\frac {1}{Y}}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y}{\partial \theta }}\right)+{\frac {1}{Y}}{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y}{\partial \varphi ^{2}}}=-\lambda .}$$

The second equation can be simplified under the assumption that Y has the form Y(θ, φ) = Θ(θ) Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations

$${\displaystyle {\frac {1}{\Phi }}{\frac {d^{2}\Phi }{d\varphi ^{2}}}=-m^{2}}$$

$${\displaystyle \lambda \sin ^{2}\theta +{\frac {\sin \theta }{\Theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)=m^{2}}$$

for some number m. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e^{± imφ}. The solution function Y(θ, φ) is regular at the poles of the sphere, where θ = 0, π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ (ℓ + 1) for some non-negative integer with ℓ ≥ |m|; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial P^m
_ℓ(cos θ) . Finally, the equation for R has solutions of the form R(r) = A r^ℓ + B r^{−ℓ − 1}; requiring the solution to be regular throughout R³ forces B = 0.[3]

Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of ℓ, there are 2ℓ + 1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. These angular solutions ${\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} }$ are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

$${\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })}$$

which fulfill

$${\displaystyle r^{2}\nabla ^{2}Y_{\ell }^{m}(\theta ,\varphi )=-\ell (\ell +1)Y_{\ell }^{m}(\theta ,\varphi ).}$$

Here ${\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} }$ is called a spherical harmonic function of degree ℓ and order m, ${\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} }$ is an associated Legendre polynomial, N is a normalization constant,[4] and θ and φ represent colatitude and longitude, respectively. In particular, the colatitude θ, or polar angle, ranges from 0 at the North Pole, to π/2 at the Equator, to π at the South Pole, and the longitude φ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer ℓ, every solution Y(θ, φ), ${\displaystyle Y:S^{2}\to \mathbb {C} }$, of the eigenvalue problem

$${\displaystyle r^{2}\nabla ^{2}Y=-\ell (\ell +1)Y}$$

is a linear combination of ${\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} }$. In fact, for any such solution, r^ℓ Y(θ, φ) is the expression in spherical coordinates of a homogeneous polynomial ${\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }$ that is harmonic (see below), and so counting dimensions shows that there are 2ℓ + 1 linearly independent such polynomials.

The general solution ${\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} }$ to Laplace's equation ${\displaystyle \Delta f=0}$ in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r^ℓ,

$${\displaystyle f(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),}$$

where the ${\displaystyle f_{\ell }^{m}\in \mathbb {C} }$ are constants and the factors r^ℓ Y_ℓ^m are known as (regular) solid harmonics ${\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }$. Such an expansion is valid in the ball

$${\displaystyle r<R={\frac {1}{\limsup _{\ell \to \infty }|f_{\ell }^{m}|^{{1}/{\ell }}}}.}$$

For ${\displaystyle r>R}$, the solid harmonics with negative powers of ${\displaystyle r}$ (the irregular solid harmonics ${\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} }$) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about ${\displaystyle r=\infty }$), instead of the Taylor series (about ${\displaystyle r=0}$) used above, to match the terms and find series expansion coefficients ${\displaystyle f_{\ell }^{m}\in \mathbb {C} }$.

Orbital angular momentum

In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[5]

$${\displaystyle \mathbf {L} =-i\hbar (\mathbf {x} \times \mathbf {\nabla } )=L_{x}\mathbf {i} +L_{y}\mathbf {j} +L_{z}\mathbf {k} .}$$

The ħ is conventional in quantum mechanics; it is convenient to work in units in which ħ = 1. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum

$${\displaystyle {\begin{aligned}\mathbf {L} ^{2}&=-r^{2}\nabla ^{2}+\left(r{\frac {\partial }{\partial r}}+1\right)r{\frac {\partial }{\partial r}}\\&=-{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial }{\partial \theta }}-{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}.\end{aligned}}}$$

Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:

$${\displaystyle {\begin{aligned}L_{z}&=-i\left(x{\frac {\partial }{\partial y}}-y{\frac {\partial }{\partial x}}\right)\\&=-i{\frac {\partial }{\partial \varphi }}.\end{aligned}}}$$

These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R³:

$${\displaystyle {\frac {1}{(2\pi )^{3/2}}}\int _{\mathbb {R} ^{3}}|f(x)|^{2}e^{-|x|^{2}/2}\,dx<\infty .}$$

Furthermore, L² is a positive operator.

If Y is a joint eigenfunction of L² and L_z, then by definition

$${\displaystyle {\begin{aligned}\mathbf {L} ^{2}Y&=\lambda Y\\L_{z}Y&=mY\end{aligned}}}$$

for some real numbers m and λ. Here m must in fact be an integer, for Y must be periodic in the coordinate φ with period a number that evenly divides 2π. Furthermore, since

$${\displaystyle \mathbf {L} ^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}}$$

and each of L_x, L_y, L_z are self-adjoint, it follows that λ ≥ m².

Denote this joint eigenspace by E_λ,m, and define the raising and lowering operators by

$${\displaystyle {\begin{aligned}L_{+}&=L_{x}+iL_{y}\\L_{-}&=L_{x}-iL_{y}\end{aligned}}}$$

Then L₊ and L₋ commute with L², and the Lie algebra generated by L₊, L₋, L_z is the special linear Lie algebra of order 2, ${\displaystyle {\mathfrak {sl}}_{2}(\mathbb {C} )}$, with commutation relations

$${\displaystyle [L_{z},L_{+}]=L_{+},\quad [L_{z},L_{-}]=-L_{-},\quad [L_{+},L_{-}]=2L_{z}.}$$

Thus L₊ : E_λ,m → E_λ,m+1 (it is a "raising operator") and L₋ : E_λ,m → E_λ,m−1 (it is a "lowering operator"). In particular, L^k
₊ : E_λ,m → E_λ,m+k must be zero for k sufficiently large, because the inequality λ ≥ m² must hold in each of the nontrivial joint eigenspaces. Let Y ∈ E_λ,m be a nonzero joint eigenfunction, and let k be the least integer such that

$${\displaystyle L_{+}^{k}Y=0.}$$

Then, since

$${\displaystyle L_{-}L_{+}=\mathbf {L} ^{2}-L_{z}^{2}-L_{z}}$$

it follows that

$${\displaystyle 0=L_{-}L_{+}^{k}Y=(\lambda -(m+k)^{2}-(m+k))Y.}$$

Thus λ = ℓ(ℓ + 1) for the positive integer ℓ = m + k.

The foregoing has been all worked out in the spherical coordinate representation, ${\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )}$ but may be expressed more abstractly in the complete, orthonormal spherical ket basis.

The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions ${\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }$. Specifically, we say that a (complex-valued) polynomial function ${\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} }$ is homogeneous of degree ${\displaystyle \ell }$ if

$${\displaystyle p(\lambda \mathbf {x} )=\lambda ^{\ell }p(\mathbf {x} )}$$

for all real numbers ${\displaystyle \lambda \in \mathbb {R} }$ and all ${\displaystyle x\in \mathbb {R} ^{3}}$. We say that ${\displaystyle p}$ is harmonic if

$${\displaystyle \Delta p=0,}$$

where ${\displaystyle \Delta }$ is the Laplacian. Then for each ${\displaystyle \ell }$, we define

$${\displaystyle \mathbf {A} _{\ell }=\left\{{\text{harmonic polynomials }}\mathbb {R} ^{3}\to \mathbb {C} {\text{ that are homogeneous of degree }}\ell \right\}.}$$

For example, when ${\displaystyle \ell =1}$, ${\displaystyle \mathbf {A} _{1}}$ is just the 3-dimensional space of all linear functions ${\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }$, since any such function is automatically harmonic. Meanwhile, when ${\displaystyle \ell =2}$, we have a 5-dimensional space:

$${\displaystyle \mathbf {A} _{2}=\operatorname {span} _{\mathbb {C} }(x_{1}x_{2},\,x_{1}x_{3},\,x_{2}x_{3},\,x_{1}^{2}-x_{2}^{2},\,x_{1}^{2}-x_{3}^{2}).}$$

For any ${\displaystyle \ell }$, the space ${\displaystyle \mathbf {H} _{\ell }}$ of spherical harmonics of degree ${\displaystyle \ell }$ is just the space of restrictions to the sphere ${\displaystyle S^{2}}$ of the elements of ${\displaystyle \mathbf {A} _{\ell }}$.[6] As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a harmonic function).

For example, for any ${\displaystyle c\in \mathbb {C} }$ the formula

$${\displaystyle p(x_{1},x_{2},x_{3})=c(x_{1}+ix_{2})^{\ell }}$$

defines a homogeneous polynomial of degree ${\displaystyle \ell }$ with domain and codomain ${\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }$, which happens to be independent of ${\displaystyle x_{3}}$. This polynomial is easily seen to be harmonic. If we write ${\displaystyle p}$ in spherical coordinates ${\displaystyle (r,\theta ,\varphi )}$ and then restrict to ${\displaystyle r=1}$, we obtain

$${\displaystyle p(\theta ,\varphi )=c\sin(\theta )^{\ell }(\cos(\varphi )+i\sin(\varphi ))^{\ell },}$$

which can be rewritten as

$${\displaystyle p(\theta ,\varphi )=c\left({\sqrt {1-\cos ^{2}(\theta )}}\right)^{\ell }e^{i\ell \varphi }.}$$

After using the formula for the associated Legendre polynomial ${\displaystyle P_{\ell }^{\ell }}$, we may recognize this as the formula for the spherical harmonic ${\displaystyle Y_{\ell }^{\ell }(\theta ,\varphi ).}$[7] (See the section below on special cases of the spherical harmonics.)

The complex spherical harmonics ${\displaystyle Y_{\ell }^{m}}$ give rise to the solid harmonics by extending from ${\displaystyle S^{2}}$ to all of ${\displaystyle \mathbb {R} ^{3}}$ as a homogeneous function of degree ${\displaystyle \ell }$, i.e. setting

$${\displaystyle R_{\ell }^{m}(v):=\|v\|^{\ell }Y_{\ell }^{m}\left({\frac {v}{\|v\|}}\right)}$$

It turns out that ${\displaystyle R_{\ell }^{m}}$ is basis of the space of harmonic and homogeneous polynomials of degree ${\displaystyle \ell }$. More specifically, it is the (unique up to normalization) Gelfand-Tsetlin-basis of this representation of the rotational group ${\displaystyle SO(3)}$ and an explicit formula for ${\displaystyle R_{\ell }^{m}}$ in cartesian coordinates can be derived from that fact.

The Herglotz generating function

If the quantum mechanical convention is adopted for the ${\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} }$, then

$${\displaystyle e^{v{\mathbf {a} }\cdot {\mathbf {r} }}=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}{\frac {r^{\ell }v^{\ell }{\lambda ^{m}}}{\sqrt {(\ell +m)!(\ell -m)!}}}Y_{\ell }^{m}(\mathbf {r} /r).}$$

Here, ${\displaystyle \mathbf {r} }$ is the vector with components ${\displaystyle (x,y,z)\in \mathbb {R} ^{3}}$, ${\displaystyle r=|\mathbf {r} |}$, and

$${\displaystyle {\mathbf {a} }={\mathbf {\hat {z}} }-{\frac {\lambda }{2}}\left({\mathbf {\hat {x}} }+i{\mathbf {\hat {y}} }\right)+{\frac {1}{2\lambda }}\left({\mathbf {\hat {x}} }-i{\mathbf {\hat {y}} }\right).}$$

${\displaystyle \mathbf {a} }$ is a vector with complex coordinates:

${\displaystyle \mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].}$

The essential property of ${\displaystyle \mathbf {a} }$ is that it is null:

$${\displaystyle \mathbf {a} \cdot \mathbf {a} =0.}$$

It suffices to take ${\displaystyle v}$ and ${\displaystyle \lambda }$ as real parameters. In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, §VII.7, who credit unpublished notes by him for its discovery.

Essentially all the properties of the spherical harmonics can be derived from this generating function.[15] An immediate benefit of this definition is that if the vector ${\displaystyle \mathbf {r} }$ is replaced by the quantum mechanical spin vector operator ${\displaystyle \mathbf {J} }$, such that ${\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })}$ is the operator analogue of the solid harmonic ${\displaystyle r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)}$,[16] one obtains a generating function for a standardized set of spherical tensor operators, ${\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })}$:

$${\displaystyle e^{v{\mathbf {a} }\cdot {\mathbf {J} }}=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}{\frac {v^{\ell }{\lambda ^{m}}}{\sqrt {(\ell +m)!(\ell -m)!}}}{\mathcal {Y}}_{\ell }^{m}({\mathbf {J} }).}$$

The parallelism of the two definitions ensures that the ${\displaystyle {\mathcal {Y}}_{\ell }^{m}}$'s transform under rotations (see below) in the same way as the ${\displaystyle Y_{\ell }^{m}}$'s, which in turn guarantees that they are spherical tensor operators, ${\displaystyle T_{q}^{(k)}}$, with ${\displaystyle k={\ell }}$ and ${\displaystyle q=m}$, obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. They are, moreover, a standardized set with a fixed scale or normalization.

See also: Spherical basis

Separated Cartesian form

The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of ${\displaystyle z}$ and another of ${\displaystyle x}$ and ${\displaystyle y}$, as follows (Condon–Shortley phase):

$${\displaystyle r^{\ell }\,{\begin{pmatrix}Y_{\ell }^{m}\\Y_{\ell }^{-m}\end{pmatrix}}=\left[{\frac {2\ell +1}{4\pi }}\right]^{1/2}{\bar {\Pi }}_{\ell }^{m}(z){\begin{pmatrix}\left(-1\right)^{m}(A_{m}+iB_{m})\\(A_{m}-iB_{m})\end{pmatrix}},\qquad m>0.}$$

and for m = 0:

$${\displaystyle r^{\ell }\,Y_{\ell }^{0}\equiv {\sqrt {\frac {2\ell +1}{4\pi }}}{\bar {\Pi }}_{\ell }^{0}.}$$

Here

$${\displaystyle A_{m}(x,y)=\sum _{p=0}^{m}{\binom {m}{p}}x^{p}y^{m-p}\cos \left((m-p){\frac {\pi }{2}}\right),}$$

$${\displaystyle B_{m}(x,y)=\sum _{p=0}^{m}{\binom {m}{p}}x^{p}y^{m-p}\sin \left((m-p){\frac {\pi }{2}}\right),}$$

and

$${\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)=\left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\sum _{k=0}^{\left\lfloor (\ell -m)/2\right\rfloor }(-1)^{k}2^{-\ell }{\binom {\ell }{k}}{\binom {2\ell -2k}{\ell }}{\frac {(\ell -2k)!}{(\ell -2k-m)!}}\;r^{2k}\;z^{\ell -2k-m}.}$$

For ${\displaystyle m=0}$ this reduces to

$${\displaystyle {\bar {\Pi }}_{\ell }^{0}(z)=\sum _{k=0}^{\left\lfloor \ell /2\right\rfloor }(-1)^{k}2^{-\ell }{\binom {\ell }{k}}{\binom {2\ell -2k}{\ell }}\;r^{2k}\;z^{\ell -2k}.}$$

The factor ${\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)}$ is essentially the associated Legendre polynomial ${\displaystyle P_{\ell }^{m}(\cos \theta )}$, and the factors ${\displaystyle (A_{m}\pm iB_{m})}$ are essentially ${\displaystyle e^{\pm im\varphi }}$.

Examples

Using the expressions for ${\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)}$, ${\displaystyle A_{m}(x,y)}$, and ${\displaystyle B_{m}(x,y)}$ listed explicitly above we obtain:

$${\displaystyle Y_{3}^{1}=-{\frac {1}{r^{3}}}\left[{\tfrac {7}{4\pi }}\cdot {\tfrac {3}{16}}\right]^{1/2}\left(5z^{2}-r^{2}\right)\left(x+iy\right)=-\left[{\tfrac {7}{4\pi }}\cdot {\tfrac {3}{16}}\right]^{1/2}\left(5\cos ^{2}\theta -1\right)\left(\sin \theta e^{i\varphi }\right)}$$

$${\displaystyle Y_{4}^{-2}={\frac {1}{r^{4}}}\left[{\tfrac {9}{4\pi }}\cdot {\tfrac {5}{32}}\right]^{1/2}\left(7z^{2}-r^{2}\right)\left(x-iy\right)^{2}=\left[{\tfrac {9}{4\pi }}\cdot {\tfrac {5}{32}}\right]^{1/2}\left(7\cos ^{2}\theta -1\right)\left(\sin ^{2}\theta e^{-2i\varphi }\right)}$$

It may be verified that this agrees with the function listed here and here.

Real forms

Using the equations above to form the real spherical harmonics, it is seen that for ${\displaystyle m>0}$ only the ${\displaystyle A_{m}}$ terms (cosines) are included, and for ${\displaystyle m<0}$ only the ${\displaystyle B_{m}}$ terms (sines) are included:

$${\displaystyle r^{\ell }\,{\begin{pmatrix}Y_{\ell m}\\Y_{\ell -m}\end{pmatrix}}={\sqrt {\frac {2\ell +1}{2\pi }}}{\bar {\Pi }}_{\ell }^{m}(z){\begin{pmatrix}A_{m}\\B_{m}\end{pmatrix}},\qquad m>0.}$$

and for m = 0:

$${\displaystyle r^{\ell }\,Y_{\ell 0}\equiv {\sqrt {\frac {2\ell +1}{4\pi }}}{\bar {\Pi }}_{\ell }^{0}.}$$

1. When ${\displaystyle m=0}$, the spherical harmonics ${\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} }$ reduce to the ordinary Legendre polynomials:
   $${\displaystyle Y_{\ell }^{0}(\theta ,\varphi )={\sqrt {\frac {2\ell +1}{4\pi }}}P_{\ell }(\cos \theta ).}$$

   
2. When ${\displaystyle m=\pm \ell }$,
   $${\displaystyle Y_{\ell }^{\pm \ell }(\theta ,\varphi )={\frac {(\mp 1)^{\ell }}{2^{\ell }\ell$$ !}}{\sqrt {\frac {(2\ell +1)!}{4\pi }}}\sin ^{\ell }\theta \,e^{\pm i\ell \varphi },}

   
   or more simply in Cartesian coordinates,
   $${\displaystyle r^{\ell }Y_{\ell }^{\pm \ell }({\mathbf {r} })={\frac {(\mp 1)^{\ell }}{2^{\ell }\ell$$ !}}{\sqrt {\frac {(2\ell +1)!}{4\pi }}}(x\pm iy)^{\ell }.}

   
3. At the north pole, where ${\displaystyle \theta =0}$, and ${\displaystyle \varphi }$ is undefined, all spherical harmonics except those with ${\displaystyle m=0}$ vanish:
   $${\displaystyle Y_{\ell }^{m}(0,\varphi )=Y_{\ell }^{m}({\mathbf {z} })={\sqrt {\frac {2\ell +1}{4\pi }}}\delta _{m0}.}$$

   

The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.

Parity

Main article: Parity (physics)

The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. Inversion is represented by the operator ${\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )}$. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with ${\displaystyle \mathbf {r} }$ being a unit vector,

$${\displaystyle Y_{\ell }^{m}(-\mathbf {r} )=(-1)^{\ell }Y_{\ell }^{m}(\mathbf {r} ).}$$

In terms of the spherical angles, parity transforms a point with coordinates ${\displaystyle \{\theta ,\varphi \}}$ to ${\displaystyle \{\pi -\theta ,\pi +\varphi \}}$. The statement of the parity of spherical harmonics is then

$${\displaystyle Y_{\ell }^{m}(\theta ,\varphi )\to Y_{\ell }^{m}(\pi -\theta ,\pi +\varphi )=(-1)^{\ell }Y_{\ell }^{m}(\theta ,\varphi )}$$

(This can be seen as follows: The associated Legendre polynomials gives (−1)^ℓ+m and from the exponential function we have (−1)^m, giving together for the spherical harmonics a parity of (−1)^ℓ.)

Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree ℓ changes the sign by a factor of (−1)^ℓ.

Rotations

The rotation of a real spherical function with m = 0 and ℓ = 3. The coefficients are not equal to the Wigner D-matrices, since real functions are shown, but can be obtained by re-decomposing the complex functions

Consider a rotation ${\displaystyle {\mathcal {R}}}$ about the origin that sends the unit vector ${\displaystyle \mathbf {r} }$ to ${\displaystyle \mathbf {r} '}$. Under this operation, a spherical harmonic of degree ${\displaystyle \ell }$ and order ${\displaystyle m}$ transforms into a linear combination of spherical harmonics of the same degree. That is,

$${\displaystyle Y_{\ell }^{m}({\mathbf {r} }')=\sum _{m'=-\ell }^{\ell }A_{mm'}Y_{\ell }^{m'}({\mathbf {r} }),}$$

where ${\displaystyle A_{mm'}}$ is a matrix of order ${\displaystyle (2\ell +1)}$ that depends on the rotation ${\displaystyle {\mathcal {R}}}$. However, this is not the standard way of expressing this property. In the standard way one writes,

$${\displaystyle Y_{\ell }^{m}({\mathbf {r} }')=\sum _{m'=-\ell }^{\ell }[D_{mm'}^{(\ell )}({\mathcal {R}})]^{*}Y_{\ell }^{m'}({\mathbf {r} }),}$$

where ${\displaystyle D_{mm'}^{(\ell )}({\mathcal {R}})^{*}}$ is the complex conjugate of an element of the Wigner D-matrix. In particular when ${\displaystyle \mathbf {r} '}$ is a ${\displaystyle \phi _{0}}$ rotation of the azimuth we get the identity,

$${\displaystyle Y_{\ell }^{m}({\mathbf {r} }')=Y_{\ell }^{m}({\mathbf {r} })e^{im\phi _{0}}.}$$

The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The ${\displaystyle Y_{\ell }^{m}}$'s of degree ${\displaystyle \ell }$ provide a basis set of functions for the irreducible representation of the group SO(3) of dimension ${\displaystyle (2\ell +1)}$. Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.

The Laplace spherical harmonics ${\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} }$ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions ${\displaystyle L_{\mathbb {C} }^{2}(S^{2})}$. On the unit sphere ${\displaystyle S^{2}}$, any square-integrable function ${\displaystyle f:S^{2}\to \mathbb {C} }$ can thus be expanded as a linear combination of these:

$${\displaystyle f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}\,Y_{\ell }^{m}(\theta ,\varphi ).}$$

This expansion holds in the sense of mean-square convergence — convergence in L² of the sphere — which is to say that

$${\displaystyle \lim _{N\to \infty }\int _{0}^{2\pi }\int _{0}^{\pi }\left|f(\theta ,\varphi )-\sum _{\ell =0}^{N}\sum _{m=-\ell }^{\ell }f_{\ell }^{m}Y_{\ell }^{m}(\theta ,\varphi )\right|^{2}\sin \theta \,d\theta \,d\varphi =0.}$$

The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:

$${\displaystyle f_{\ell }^{m}=\int _{\Omega }f(\theta ,\varphi )\,Y_{\ell }^{m*}(\theta ,\varphi )\,d\Omega =\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }\,d\theta \,\sin \theta f(\theta ,\varphi )Y_{\ell }^{m*}(\theta ,\varphi ).}$$

If the coefficients decay in ℓ sufficiently rapidly — for instance, exponentially — then the series also converges uniformly to f.

A square-integrable function ${\displaystyle f:S^{2}\to \mathbb {R} }$ can also be expanded in terms of the real harmonics ${\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} }$ above as a sum

$${\displaystyle f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell m}\,Y_{\ell m}(\theta ,\varphi ).}$$

The convergence of the series holds again in the same sense, namely the real spherical harmonics ${\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} }$ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions ${\displaystyle L_{\mathbb {R} }^{2}(S^{2})}$. The benefit of the expansion in terms of the real harmonic functions ${\displaystyle Y_{\ell m}}$ is that for real functions ${\displaystyle f:S^{2}\to \mathbb {R} }$ the expansion coefficients ${\displaystyle f_{\ell m}}$ are guaranteed to be real, whereas their coefficients ${\displaystyle f_{\ell }^{m}}$ in their expansion in terms of the ${\displaystyle Y_{\ell }^{m}}$ (considering them as functions ${\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} }$) do not have that property.