Inmathematics andphysical science,spherical harmonics arespecial functions defined on the surface of asphere. They are often employed in solvingpartial differential equations in many scientific fields. Thetable of spherical harmonics contains a list of common spherical harmonics.

Since the spherical harmonics form a complete set oforthogonal functions and thus anorthonormal basis, every function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar toperiodic functions defined on a circle that can be expressed as a sum ofcircular functions (sines and cosines) viaFourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial)angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics arebasis functions forirreducible representations ofSO(3), thegroup of rotations in three dimensions, and thus play a central role in thegroup theoretic discussion of SO(3).

Spherical harmonics originate from solvingLaplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are calledharmonics. Despite their name, spherical harmonics take their simplest form inCartesian coordinates, where they can be defined ashomogeneous polynomials ofdegree${\displaystyle \ell }$ in${\displaystyle (x,y,z)}$ that obey Laplace's equation. The connection withspherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence${\displaystyle r^{\ell }}$ from the above-mentioned polynomial of degree${\displaystyle \ell }$; the remaining factor can be regarded as a function of the spherical angular coordinates${\displaystyle \theta }$ and${\displaystyle \phi }$ only, or equivalently of theorientationalunit vector${\displaystyle \mathbf{r}}$ specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics arenot functions on the sphere which are harmonic with respect to theLaplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy theMaximum principle. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (seeHigher dimensions).

A specific set of spherical harmonics, denoted${\displaystyle Y_{\ell }^m(\theta ,\phi )}$ or${\displaystyle Y_{\ell }^m(\mathbf{r})}$, are known as Laplace's spherical harmonics, as they were first introduced byPierre Simon de Laplace in 1782.^[1] These functions form anorthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.

Spherical harmonics are important in many theoretical and practical applications, including the representation ofmultipole electrostatic andelectromagnetic fields,electron configurations,gravitational fields,geoids, themagnetic fields of planetary bodies and stars, and thecosmic microwave background radiation. In3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion,global illumination,precomputed radiance transfer, etc.) and modelling of 3D shapes.

Spherical harmonics were first investigated in connection with theNewtonian potential ofNewton's law of universal gravitation in three dimensions. In 1782,Pierre-Simon de Laplace had, in hisMécanique Céleste, determined that thegravitational potential${\displaystyle {\mathbb{R}}^3\to \mathbb{R}}$  at a pointx associated with a set of point massesm_i located at pointsx_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 ofr = |x| andr₁ = |x₁|. He discovered that ifr ≤r₁ then

$${\displaystyle \frac{1}{|{\mathbf{x}}_1-\mathbf{x}|}=P_0(\cos \gamma )\frac{1}{r_1}+P_1(\cos \gamma )\frac{r}{r_1^2}+P_2(\cos \gamma )\frac{r^2}{r_1^3}+\cdots }$$ 

whereγ is the angle between the vectorsx andx₁. The functions${\displaystyle P_i:[-1,1]\to \mathbb{R}}$  are theLegendre 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 angleγ betweenx₁ andx. (SeeLegendre polynomials § Applications for more detail.)

In 1867,William Thomson (Lord Kelvin) andPeter Guthrie Tait introduced thesolid spherical harmonics in theirTreatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. Thesolid harmonics werehomogeneous polynomial solutions${\displaystyle {\mathbb{R}}^3\to \mathbb{R}}$  ofLaplace's equation$${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}=0.}$$ By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. (SeeHarmonic polynomial representation.) The term "Laplace's coefficients" was employed byWilliam Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for thezonal spherical harmonics that had properly been introduced by Laplace and Legendre.

The 19th century development ofFourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of theheat equation andwave equation. This could be achieved by expansion of functions in series oftrigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in astring, the spherical harmonics represent the fundamental modes ofvibration 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 ascomplex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. This was a boon for problems possessingspherical 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 ofquantum mechanics. The (complex-valued) spherical harmonics${\displaystyle S^2\to \mathbb{C}}$  areeigenfunctions of the square of theorbital angular momentum operator$${\displaystyle -i\hslash \mathbf{r}\times \mathrm{\nabla },}$$ and therefore they represent the differentquantized configurations ofatomic orbitals.

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

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

Consider the problem of finding solutions of the formf(r,θ,φ) =R(r)Y(θ,φ). Byseparation of variables, two differential equations result by imposing Laplace's equation:$${\displaystyle \frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)=\lambda ,\frac{1}{Y}\frac{1}{\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }Y}{\mathrm{\partial }\theta }\right)+\frac{1}{Y}\frac{1}{{\sin }^2\theta }\frac{{\mathrm{\partial }}^{2}Y}{\mathrm{\partial }{\phi }^2}=-\lambda .}$$ The second equation can be simplified under the assumption thatY has the formY(θ,φ) = Θ(θ) Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations

$${\displaystyle \frac{1}{\mathrm{\Phi }}\frac{d^2\mathrm{\Phi }}{d{\phi }^2}=-m^2}$$ $${\displaystyle \lambda {\sin }^2\theta +\frac{\sin \theta }{\mathrm{\Theta }}\frac{d}{d\theta }\left(\sin \theta \frac{d\mathrm{\Theta }}{d\theta }\right)=m^2}$$ 

for some numberm. A priori,m is a complex constant, but becauseΦ must be aperiodic function whose period evenly divides2π,m is necessarily an integer andΦ is a linear combination of the complex exponentialse^±imφ. The solution functionY(θ,φ) is regular at the poles of the sphere, whereθ = 0,π. Imposing this regularity in the solutionΘ of the second equation at the boundary points of the domain is aSturm–Liouville problem that forces the parameterλ to be of the formλ =ℓ (ℓ + 1) for some non-negative integer withℓ ≥ |m|; this is also explainedbelow in terms of theorbital angular momentum. Furthermore, a change of variablest = cosθ transforms this equation into theLegendre equation, whose solution is a multiple of theassociated Legendre polynomialP^m
_ℓ(cosθ) . Finally, the equation forR has solutions of the formR(r) =A r^ℓ +B r^{−ℓ − 1}; requiring the solution to be regular throughoutR³ forcesB = 0.^[3]

Here the solution was assumed to have the special formY(θ,φ) = Θ(θ) Φ(φ). For a given value ofℓ, there are2ℓ + 1 independent solutions of this form, one for each integerm with−ℓ ≤m ≤ℓ. These angular solutions${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$  are a product oftrigonometric functions, here represented as acomplex exponential, and associated Legendre polynomials:

$${\displaystyle Y_{\ell }^m(\theta ,\phi )=N{e}^{im\phi }{P}_{\ell }^m(\cos \theta )}$$ 

which fulfill$${\displaystyle r^2{\mathrm{\nabla }}^2{Y}_{\ell }^m(\theta ,\phi )=-\ell (\ell +1)Y_{\ell }^m(\theta ,\phi ).}$$ 

Here${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$  is called aspherical harmonic function of degreeℓ and orderm,${\displaystyle P_{\ell }^m:[-1,1]\to \mathbb{R}}$  is anassociated Legendre polynomial,N is a normalization constant,^[4] andθ andφ represent colatitude and longitude, respectively. In particular, thecolatitudeθ, or polar angle, ranges from0 at the North Pole, toπ/2 at the Equator, toπ at the South Pole, and thelongitudeφ, orazimuth, may assume all values with0 ≤φ < 2π. For a fixed integerℓ, every solutionY(θ,φ),${\displaystyle Y:S^2\to \mathbb{C}}$ , of the eigenvalue problem$${\displaystyle r^2{\mathrm{\nabla }}^{2}Y=-\ell (\ell +1)Y}$$ is alinear 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 ahomogeneous polynomial${\displaystyle {\mathbb{R}}^3\to \mathbb{C}}$  that is harmonic (seebelow), and so counting dimensions shows that there are2ℓ + 1 linearly independent such polynomials.

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

$${\displaystyle f(r,\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }{f}_{\ell }^m{r}^{\ell }{Y}_{\ell }^m(\theta ,\phi ),}$$ 

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

 

For${\displaystyle r>R}$ , the solid harmonics with negative powers of${\displaystyle r}$  (theirregularsolid 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 inLaurent series (about${\displaystyle r=\mathrm{\infty }}$ ), instead of theTaylor series (about${\displaystyle r=0}$ ) used above, to match the terms and find series expansion coefficients${\displaystyle f_{\ell }^m\in \mathbb{C}}$ .

In quantum mechanics, Laplace's spherical harmonics are understood in terms of theorbital angular momentum^[5]$${\displaystyle \mathbf{L}=-i\hslash (\mathbf{x}\times \mathrm{\nabla })=L_x\mathbf{i}+L_y\mathbf{j}+L_z\mathbf{k}.}$$ Theħ is conventional in quantum mechanics; it is convenient to work in units in whichħ = 1. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum 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: 

These operators commute, and aredensely definedself-adjoint operators on theweightedHilbert space of functionsf square-integrable with respect to thenormal distribution as the weight function onR³: Furthermore,L² is apositive operator.

IfY is a joint eigenfunction ofL² andL_z, then by definition for some real numbersm andλ. Herem must in fact be an integer, forY must be periodic in the coordinateφ 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 ofL_x,L_y,L_z are self-adjoint, it follows thatλ ≥m².

Denote this joint eigenspace byE_λ,m, and define theraising and lowering operators by ThenL₊ andL₋ commute withL², and the Lie algebra generated byL₊,L₋,L_z is thespecial linear Lie algebra of order 2,${\displaystyle {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})}$ , with commutation relations$${\displaystyle [L_z,L_{+}]=L_{+},[L_z,L_{-}]=-L_{-},[L_{+},L_{-}]=2L_z.}$$ ThusL₊ :E_λ,m →E_λ,m+1 (it is a "raising operator") andL₋ :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 fork sufficiently large, because the inequalityλ ≥m² must hold in each of the nontrivial joint eigenspaces. LetY ∈E_λ,m be a nonzero joint eigenfunction, and letk 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$${\displaystyle 0=L_{-}{L}_{+}^{k}Y=(\lambda -(m+k{)}^2-(m+k))Y.}$$ Thusλ =ℓ(ℓ + 1) for the positive integerℓ =m +k.

The foregoing has been all worked out in the spherical coordinate representation,${\displaystyle ⟨\theta ,\phi |lm⟩=Y_l^m(\theta ,\phi )}$  but may be expressed more abstractly in the complete, orthonormalspherical 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${\displaystyle p:{\mathbb{R}}^3\to \mathbb{C}}$  ishomogeneous 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${\displaystyle \mathbf{x}\in {\mathbb{R}}^3}$ . We say that${\displaystyle p}$  isharmonic if$${\displaystyle \mathrm{\Delta }p=0,}$$ where${\displaystyle \mathrm{\Delta }}$  is theLaplacian. 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 6-dimensional space:$${\displaystyle {\mathbf{A}}_2={\mathrm{span}}_{\mathbb{C}}(x_1{x}_2,x_1{x}_3,x_2{x}_3,x_1^2,x_2^2,x_3^2).}$$ 

A general formula for the dimension,${\displaystyle d_l}$ , of the set of homogenous polynomials of degree${\displaystyle \ell }$  in${\displaystyle {\mathbb{R}}^n}$  is^[6]$${\displaystyle d_l=\frac{(n+l-1)!}{(n-1)!l!}}$$ 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${\displaystyle {\mathbf{A}}_{\ell }}$ .^[7] As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of aharmonic function).

For example, for any${\displaystyle c\in \mathbb{C}}$  the formula$${\displaystyle p(x_1,x_2,x_3)=c(x_1+i{x}_2{)}^{\ell }}$$ 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${\displaystyle (r,\theta ,\phi )}$  and then restrict to${\displaystyle r=1}$ , we obtain$${\displaystyle p(\theta ,\phi )=c\sin (\theta {)}^{\ell }(\cos (\phi )+i\sin (\phi ){)}^{\ell },}$$ which can be rewritten as$${\displaystyle p(\theta ,\phi )=c{\left(\sqrt{1-{\cos }^2(\theta )}\right)}^{\ell }{e}^{i\ell \phi }.}$$ After using the formula for theassociated Legendre polynomial${\displaystyle P_{\ell }^{\ell }}$ , we may recognize this as the formula for the spherical harmonic${\displaystyle Y_{\ell }^{\ell }(\theta ,\phi ).}$ ^[8] (SeeSpecial cases.)

Several different normalizations are in common use for the Laplace spherical harmonic functions${\displaystyle S^2\to \mathbb{C}}$ . Throughout the section, we use the standard convention that for${\displaystyle m>0}$  (seeassociated Legendre polynomials)$${\displaystyle P_{\ell }^{-m}=(-1{)}^m\frac{(\ell -m)!}{(\ell +m)!}{P}_{\ell }^m}$$ which is the natural normalization given by Rodrigues' formula.

Inacoustics,^[9] the Laplace spherical harmonics are generally defined as (this is the convention used in this article)$${\displaystyle Y_{\ell }^m(\theta ,\phi )=\sqrt{\frac{(2\ell +1)}{4\pi }\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\phi }}$$ while inquantum mechanics:^[10][11]$${\displaystyle Y_{\ell }^m(\theta ,\phi )=(-1{)}^m\sqrt{\frac{(2\ell +1)}{4\pi }\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\phi }}$$ 

where${\displaystyle P_{\ell }^m}$  are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice).

In both definitions, the spherical harmonics are orthonormal$${\displaystyle {\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{Y}_{\ell }^m{Y}_{{\ell }^{\prime }}^{m^{\prime }}{}^{\ast }d\mathrm{\Omega }={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }},}$$ whereδ_ij is theKronecker delta anddΩ = sin(θ)dφdθ. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.,$${\displaystyle \int |Y_{\ell }^m{|}^{2}d\mathrm{\Omega }=1.}$$ 

The disciplines ofgeodesy^[12] and spectral analysis use

$${\displaystyle Y_{\ell }^m(\theta ,\phi )=\sqrt{(2\ell +1)\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\phi }}$$ 

which possess unit power

$${\displaystyle \frac{1}{4\pi }{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{Y}_{\ell }^m{Y}_{{\ell }^{\prime }}^{m^{\prime }}{}^{\ast }d\mathrm{\Omega }={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.}$$ 

Themagnetics^[12] community, in contrast, uses Schmidt semi-normalized harmonics

$${\displaystyle Y_{\ell }^m(\theta ,\phi )=\sqrt{\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\phi }}$$ 

which have the normalization

$${\displaystyle {\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{Y}_{\ell }^m{Y}_{{\ell }^{\prime }}^{m^{\prime }}{}^{\ast }d\mathrm{\Omega }=\frac{4\pi }{(2\ell +1)}{\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.}$$ 

In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization afterGiulio Racah.

It can be shown that all of the above normalized spherical harmonic functions satisfy

$${\displaystyle Y_{\ell }^m{}^{\ast }(\theta ,\phi )=(-1{)}^{-m}{Y}_{\ell }^{-m}(\theta ,\phi ),}$$ 

where the superscript* denotes complex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with theWigner D-matrix.

One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of${\displaystyle (-1{)}^m}$ , commonly referred to as theCondon–Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include thisphase factor in the definition of theassociated Legendre polynomials, or to append it to the definition of the spherical harmonic functions. There is no requirement to use the Condon–Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application ofraising and lowering operators. The geodesy^[13] and magnetics communities never include the Condon–Shortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials.^[14]

A real basis of spherical harmonics${\displaystyle Y_{\ell m}:S^2\to \mathbb{R}}$  can be defined in terms of their complex analogues${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$  by setting The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$  in terms of the real spherical harmonics${\displaystyle Y_{\ell m}:S^2\to \mathbb{R}}$  are 

The real spherical harmonics${\displaystyle Y_{\ell m}:S^2\to \mathbb{R}}$  are sometimes known astesseral spherical harmonics.^[15] These functions have the same orthonormality properties as the complex ones${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$  above. The real spherical harmonics${\displaystyle Y_{\ell m}}$  withm > 0 are said to be of cosine type, and those withm < 0 of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as 

The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation.

Seehere for a list of real spherical harmonics up to and including${\displaystyle \ell =4}$ , which can be seen to be consistent with the output of the equations above.

As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. However, the solutions of the non-relativistic Schrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, the real functions span the same space as the complex ones would.

For example, as can be seen from thetable of spherical harmonics, the usualp functions (${\displaystyle \ell =1}$ ) are complex and mix axis directions, but thereal versions are essentially justx,y, andz.

The complex spherical harmonics${\displaystyle Y_{\ell }^m}$  give rise to thesolid harmonics by extending from${\displaystyle S^2}$  to all of${\displaystyle {\mathbb{R}}^3}$  as ahomogeneous function of degree${\displaystyle \ell }$ , i.e. setting$${\displaystyle R_{\ell }^m(v):=\Vert v{\Vert }^{\ell }{Y}_{\ell }^m\left(\frac{v}{\Vert v\Vert }\right)}$$ It turns out that${\displaystyle R_{\ell }^m}$  is basis of the space of harmonic andhomogeneous 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 anexplicit formula for${\displaystyle R_{\ell }^m}$  in cartesian coordinates can be derived from that fact.

If the quantum mechanical convention is adopted for the${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$ , then$${\displaystyle e^{v\mathbf{a}\cdot \mathbf{r}}=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }\sqrt{\frac{4\pi }{2\ell +1}}\frac{r^{\ell }{v}^{\ell }{\lambda }^m}{\sqrt{(\ell +m)!(\ell -m)!}}{Y}_{\ell }^m(\mathbf{r}/r).}$$ Here,${\displaystyle \mathbf{r}}$  is the vector with components${\displaystyle (x,y,z)\in {\mathbb{R}}^3}$ ,${\displaystyle r=|\mathbf{r}|}$ , and$${\displaystyle \mathbf{a}=\hat{\mathbf{z}}-\frac{\lambda }{2}\left(\hat{\mathbf{x}}+i\hat{\mathbf{y}}\right)+\frac{1}{2\lambda }\left(\hat{\mathbf{x}}-i\hat{\mathbf{y}}\right).}$$ ${\displaystyle \mathbf{a}}$  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 afterHerglotz, we followCourant & 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.^[16] 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 thesolid harmonic${\displaystyle r^{\ell }{Y}_{\ell }^m(\mathbf{r}/r)}$ ,^[17] one obtains a generating function for a standardized set ofspherical tensor operators,${\displaystyle {\mathcal{Y}}_{\ell }^m(\mathbf{J})}$ :

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

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 theClebsch-Gordan composition theorem, and theWigner-Eckart theorem. They are, moreover, a standardized set with a fixed scale or normalization.

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):$${\displaystyle r^{\ell }\left(\begin{array}{c}{Y}_{\ell }^m\\ Y_{\ell }^{-m}\end{array}\right)={\left[\frac{2\ell +1}{4\pi }\right]}^{1/2}{\overline{\mathrm{\Pi }}}_{\ell }^m(z)\left(\begin{array}{c}{\left(-1\right)}^m(A_m+i{B}_m)\\ (A_m-i{B}_m)\end{array}\right),m>0.}$$ and form = 0:$${\displaystyle r^{\ell }{Y}_{\ell }^0\equiv \sqrt{\frac{2\ell +1}{4\pi }}{\overline{\mathrm{\Pi }}}_{\ell }^0.}$$ Here$${\displaystyle A_m(x,y)=\sum _{p=0}^m(\genfrac{}{}{0ex}{}{m}{p})x^p{y}^{m-p}\cos \left((m-p)\frac{\pi }{2}\right),}$$ $${\displaystyle B_m(x,y)=\sum _{p=0}^m(\genfrac{}{}{0ex}{}{m}{p})x^p{y}^{m-p}\sin \left((m-p)\frac{\pi }{2}\right),}$$ and$${\displaystyle {\overline{\mathrm{\Pi }}}_{\ell }^m(z)={\left[\frac{(\ell -m)!}{(\ell +m)!}\right]}^{1/2}\sum _{k=0}^{\lfloor (\ell -m)/2\rfloor }(-1{)}^k{2}^{-\ell }(\genfrac{}{}{0ex}{}{\ell }{k})(\genfrac{}{}{0ex}{}{2\ell -2k}{\ell })\frac{(\ell -2k)!}{(\ell -2k-m)!}{r}^{2k}{z}^{\ell -2k-m}.}$$ For${\displaystyle m=0}$  this reduces to$${\displaystyle {\overline{\mathrm{\Pi }}}_{\ell }^0(z)=\sum _{k=0}^{\lfloor \ell /2\rfloor }(-1{)}^k{2}^{-\ell }(\genfrac{}{}{0ex}{}{\ell }{k})(\genfrac{}{}{0ex}{}{2\ell -2k}{\ell })r^{2k}{z}^{\ell -2k}.}$$ 

The factor${\displaystyle {\overline{\mathrm{\Pi }}}_{\ell }^m(z)}$  is essentially the associated Legendre polynomial${\displaystyle P_{\ell }^m(\cos \theta )}$ , and the factors${\displaystyle (A_m\pm i{B}_m)}$  are essentially${\displaystyle e^{\pm im\phi }}$ .

Using the expressions for${\displaystyle {\overline{\mathrm{\Pi }}}_{\ell }^m(z)}$ ,${\displaystyle A_m(x,y)}$ , and${\displaystyle B_m(x,y)}$  listed explicitly above we obtain:$${\displaystyle Y_3^1=-\frac{1}{r^3}{\left[\frac{7}{4\pi }\cdot \frac{3}{16}\right]}^{1/2}\left(5z^2-r^2\right)\left(x+iy\right)=-{\left[\frac{7}{4\pi }\cdot \frac{3}{16}\right]}^{1/2}\left(5{\cos }^2\theta -1\right)\left(\sin \theta e^{i\phi }\right)}$$ 

$${\displaystyle Y_4^{-2}=\frac{1}{r^4}{\left[\frac{9}{4\pi }\cdot \frac{5}{32}\right]}^{1/2}\left(7z^2-r^2\right){\left(x-iy\right)}^2={\left[\frac{9}{4\pi }\cdot \frac{5}{32}\right]}^{1/2}\left(7{\cos }^2\theta -1\right)\left({\sin }^2\theta e^{-2i\phi }\right)}$$ It may be verified that this agrees with the function listedhere andhere.

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  only the${\displaystyle B_m}$  terms (sines) are included:

$${\displaystyle r^{\ell }\left(\begin{array}{c}{Y}_{\ell m}\\ Y_{\ell -m}\end{array}\right)=\sqrt{\frac{2\ell +1}{2\pi }}{\overline{\mathrm{\Pi }}}_{\ell }^m(z)\left(\begin{array}{c}{A}_m\\ B_m\end{array}\right),m>0.}$$ and form = 0:$${\displaystyle r^{\ell }{Y}_{\ell 0}\equiv \sqrt{\frac{2\ell +1}{4\pi }}{\overline{\mathrm{\Pi }}}_{\ell }^0.}$$ 

1. When${\displaystyle m=0}$ , the spherical harmonics${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$  reduce to the ordinaryLegendre polynomials:$${\displaystyle Y_{\ell }^0(\theta ,\phi )=\sqrt{\frac{2\ell +1}{4\pi }}{P}_{\ell }(\cos \theta ).}$$ 
2. When${\displaystyle m=\pm \ell }$ ,$${\displaystyle Y_{\ell }^{\pm \ell }(\theta ,\phi )=\frac{(\mp 1{)}^{\ell }}{2^{\ell }\ell !}\sqrt{\frac{(2\ell +1)!}{4\pi }}{\sin }^{\ell }\theta e^{\pm i\ell \phi },}$$  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${\displaystyle \phi }$  is undefined, all spherical harmonics except those with${\displaystyle m=0}$  vanish:$${\displaystyle Y_{\ell }^m(0,\phi )=Y_{\ell }^m(\mathbf{z})=\sqrt{\frac{2\ell +1}{4\pi }}{\delta }_{m0}.}$$ 

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

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${\displaystyle P\mathrm{\Psi }(\mathbf{r})=\mathrm{\Psi }(-\mathbf{r})}$ . 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${\displaystyle \{\theta ,\phi \}}$  to${\displaystyle \{\pi -\theta ,\pi +\phi \}}$ . The statement of the parity of spherical harmonics is then$${\displaystyle Y_{\ell }^m(\theta ,\phi )\to Y_{\ell }^m(\pi -\theta ,\pi +\phi )=(-1{)}^{\ell }{Y}_{\ell }^m(\theta ,\phi )}$$ (This can be seen as follows: Theassociated 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 apoint reflection to a spherical harmonic of degreeℓ changes the sign by a factor of(−1)^ℓ.

Consider a rotation${\displaystyle \mathcal{R}}$  about the origin that sends the unit vector${\displaystyle \mathbf{r}}$  to${\displaystyle {\mathbf{r}}^{\prime }}$ . 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,$${\displaystyle Y_{\ell }^m({\mathbf{r}}^{\prime })=\sum _{m^{\prime }=-\ell }^{\ell }{A}_{m{m}^{\prime }}{Y}_{\ell }^{m^{\prime }}(\mathbf{r}),}$$ where${\displaystyle A_{m{m}^{\prime }}}$  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,

$${\displaystyle Y_{\ell }^m({\mathbf{r}}^{\prime })=\sum _{m^{\prime }=-\ell }^{\ell }[D_{m{m}^{\prime }}^{(\ell )}(\mathcal{R}){]}^{\ast }{Y}_{\ell }^{m^{\prime }}(\mathbf{r}),}$$ where${\displaystyle D_{m{m}^{\prime }}^{(\ell )}(\mathcal{R}{)}^{\ast }}$  is the complex conjugate of an element of theWigner D-matrix. In particular when${\displaystyle {\mathbf{r}}^{\prime }}$  is a${\displaystyle {\varphi }_0}$  rotation of the azimuth we get the identity,

$${\displaystyle Y_{\ell }^m({\mathbf{r}}^{\prime })=Y_{\ell }^m(\mathbf{r})e^{im{\varphi }_0}.}$$ 

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 anorthonormal basis of theHilbert space ofsquare-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:

$${\displaystyle f(\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }{f}_{\ell }^m{Y}_{\ell }^m(\theta ,\phi ).}$$ 

This expansion holds in the sense of mean-square convergence — convergence inL² of the sphere — which is to say that

$${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\int }_0^{2\pi }{\int }_0^{\pi }{\left|f(\theta ,\phi )-\sum _{\ell =0}^N\sum _{m=-\ell }^{\ell }{f}_{\ell }^m{Y}_{\ell }^m(\theta ,\phi )\right|}^2\sin \theta d\theta d\phi =0.}$$ 

The expansion coefficients are the analogs ofFourier 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:

$${\displaystyle f_{\ell }^m={\int }_{\mathrm{\Omega }}f(\theta ,\phi )Y_{\ell }^{m\ast }(\theta ,\phi )d\mathrm{\Omega }={\int }_0^{2\pi }d\phi {\int }_0^{\pi }d\theta \sin \theta f(\theta ,\phi )Y_{\ell }^{m\ast }(\theta ,\phi ).}$$ 

If the coefficients decay inℓ sufficiently rapidly — for instance,exponentially — then the series alsoconverges uniformly tof.

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

$${\displaystyle f(\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }{f}_{\ell m}{Y}_{\ell m}(\theta ,\phi ).}$$ 

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 anorthonormal basis of theHilbert space ofsquare-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.

The total power of a functionf is defined in thesignal processing literature as the integral of the function squared, divided by the area of its domain. Using theorthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization ofParseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics):

$${\displaystyle \frac{1}{4\pi }{\int }_{\mathrm{\Omega }}|f(\mathrm{\Omega }){|}^{2}d\mathrm{\Omega }=\sum _{\ell =0}^{\mathrm{\infty }}{S}_{ff}(\ell ),}$$ where$${\displaystyle S_{ff}(\ell )=\frac{1}{2\ell +1}\sum _{m=-\ell }^{\ell }|f_{\ell m}{|}^2}$$ 

is defined as the angular power spectrum (for Schmidt semi-normalized harmonics). In a similar manner, one can define the cross-power of two functions as$${\displaystyle \frac{1}{4\pi }{\int }_{\mathrm{\Omega }}f(\mathrm{\Omega })g^{\ast }(\mathrm{\Omega })d\mathrm{\Omega }=\sum _{\ell =0}^{\mathrm{\infty }}{S}_{fg}(\ell ),}$$ where$${\displaystyle S_{fg}(\ell )=\frac{1}{2\ell +1}\sum _{m=-\ell }^{\ell }{f}_{\ell m}{g}_{\ell m}^{\ast }}$$