View source for Physics:Spherical harmonics

{{short description|Special mathematical functions defined on the surface of a sphere}}
[[Image:Spherical Harmonics.png|right|thumb|300px|Visual representations of the first few real spherical harmonics. Blue portions represent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from the origin indicates the absolute value of <math>Y_\ell^m(\theta,\varphi)</math> in angular direction <math>(\theta,\varphi)</math>.]]

In [[Mathematics|mathematics]] and [[Physics:Outline of physical science|physical science]], '''spherical harmonics''' are special functions defined on the surface of a [[Sphere|sphere]]. They are often employed in solving [[Partial differential equation|partial differential equation]]s in many scientific fields. A list of the spherical harmonics is available in [[Table of spherical harmonics]].

Since the spherical harmonics form a complete set of [[Orthogonal functions|orthogonal functions]] and thus an [[Orthonormal basis|orthonormal basis]], each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to [[Periodic function|periodic function]]s defined on a circle that can be expressed as a sum of [[Trigonometric functions|circular functions]] (sines and cosines) via [[Fourier series]]. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) [[Physics:Angular frequency|angular frequency]], as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are [[Basis function|basis function]]s for irreducible representations of [[Rotation group SO(3)|SO(3)]], the [[Group (mathematics)|group]] of rotations in three dimensions, and thus play a central role in the [[Group theory|group theoretic]] discussion of SO(3).

Spherical harmonics originate from solving [[Laplace's equation]] in the spherical domains. Functions that are solutions to Laplace's equation are called [[Harmonic function|harmonics]]. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as [[Homogeneous polynomial|homogeneous polynomial]]s of [[Degree of a polynomial|degree]] <math>\ell</math> in <math>(x, y, z)</math> that obey Laplace's equation. The connection with [[Spherical coordinate system|spherical coordinates]] arises immediately if one uses the homogeneity to extract a factor of radial dependence <math>r^\ell</math> from the above-mentioned polynomial of degree <math>\ell</math>; the remaining factor can be regarded as a function of the spherical angular coordinates <math>\theta</math> and <math>\varphi</math> only, or equivalently of the [[Orientation (geometry)|orientational]] [[Unit vector|unit vector]] <math>\mathbf r</math> 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 are ''not'' functions on the sphere which are harmonic with respect to the Laplace-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 the [[Maximum principle]]. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section ''Higher dimensions'' below).

A specific set of spherical harmonics, denoted <math>Y_\ell^m(\theta,\varphi)</math> or <math>Y_\ell^m({\mathbf r})</math>, are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782.<ref>A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of {{harvnb|MacRobert|1967}}. The term "Laplace spherical harmonics" is in common use; see {{harvnb|Courant|Hilbert|1962}} and {{harvnb|Meijer|Bauer|2004}}.</ref> These functions form an orthogonal 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 of [[Multipole expansion|multipole]] electrostatic and [[Physics:Electromagnetic field|electromagnetic field]]s, [[Physics:Electron configuration|electron configuration]]s, [[Physics:Gravitational field|gravitational field]]s, [[Geoid|geoid]]s, the [[Physics:Magnetic field|magnetic field]]s of planetary bodies and stars, and the cosmic microwave background radiation. In [[3D computer graphics|3D computer graphics]], spherical harmonics play a role in a wide variety of topics including indirect lighting ([[Ambient occlusion|ambient occlusion]], [[Global illumination|global illumination]], [[Precomputed Radiance Transfer|precomputed radiance transfer]], etc.) and modelling of 3D shapes.

==History==
[[File:Laplace, Pierre-Simon, marquis de.jpg|thumb|[[Biography:Pierre-Simon Laplace|Pierre-Simon Laplace]],  1749–1827]]
Spherical harmonics were first investigated in connection with the [[Newtonian potential]] of [[Physics:Newton's law of universal gravitation|Newton's law of universal gravitation]] in three dimensions. In 1782, Pierre-Simon de Laplace had, in his ''Mécanique Céleste'', determined that the [[Physics:Gravitational potential|gravitational potential]] <math>\R^3 \to \R</math> at a point {{math|'''x'''}} associated with a set of point masses {{math|''m''<sub>''i''</sub>}} located at points {{math|'''x'''<sub>''i''</sub>}} was given by

<math display="block">V(\mathbf{x}) = \sum_i \frac{m_i}{|\mathbf{x}_i - \mathbf{x}|}.</math>

Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time, [[Biography:Adrien-Marie Legendre|Adrien-Marie Legendre]] had investigated the expansion of the Newtonian potential in powers of {{math|1=''r'' = {{abs|'''x'''}}}} and {{math|1=''r''<sub>1</sub> = {{abs|'''x'''<sub>1</sub>}}}}. He discovered that if {{math|''r'' ≤ ''r''<sub>1</sub>}} then

<math display="block">\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</math>

where {{math|''γ''}} is the angle between the vectors {{math|'''x'''}} and {{math|'''x'''<sub>1</sub>}}. The functions <math>P_i: [-1, 1] \to \R</math> 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 {{math|''γ''}} between {{math|'''x'''<sub>1</sub>}} and {{math|'''x'''}}. (See [[Legendre polynomials#Applications of Legendre polynomials in physics|Applications of Legendre polynomials in physics]] for a more detailed analysis.)

In 1867, [[Biography:William Thomson, 1st Baron Kelvin|William Thomson]] (Lord Kelvin) and Peter Guthrie Tait introduced the [[Physics:Solid harmonics|solid spherical harmonics]] in their ''[[Physics:Treatise on Natural Philosophy|Treatise on Natural Philosophy]]'', and also first introduced the name of "spherical harmonics" for these functions. The [[Physics:Solid harmonics|solid harmonics]] were [[Homogeneous function|homogeneous]] polynomial solutions <math>\R^3 \to \R</math> of [[Laplace's equation]]
<math display="block">\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0.</math>
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 [[Biography:William Whewell|William Whewell]] to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the [[Zonal spherical harmonics|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|heat equation]] and [[Physics:Wave equation|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 [[Physics:Wave equation#Spherical waves|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 [[Euler's formula|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 [[Physics:Quantum mechanics|quantum mechanics]]. The (complex-valued) spherical harmonics <math>S^2 \to \Complex</math> are [[Eigenfunction|eigenfunction]]s of the square of the [[Physics:Angular momentum operator|orbital angular momentum]] operator
<math display="block">-i\hbar\mathbf{r}\times\nabla,</math>
and therefore they represent the different quantized configurations of atomic orbitals.

==Laplace's spherical harmonics==
[[File:Rotating spherical harmonics.gif|thumb|right|Real (Laplace) spherical harmonics <math>Y_{\ell m}</math> for <math>\ell=0,\dots,4</math> (top to bottom) and <math>m=0,\dots,\ell</math> (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 <math>Y_{\ell(-m)}</math> would be shown rotated about the ''z'' axis by <math>90^\circ/m</math> with respect to the positive order ones.)]]

[[File:Sphericalfunctions.svg|thumb|500px|Alternative picture for the real spherical harmonics <math>Y_{\ell m}</math>.]]

[[Laplace's equation]] imposes that the Laplacian of a scalar field {{math|''f''}} is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function <math>f:\R^3 \to \Complex</math>.) In [[Spherical coordinate system|spherical coordinates]] this is:<ref>The approach to spherical harmonics taken here is found in {{harv|Courant|Hilbert|1962|loc=§V.8, §VII.5}}.</ref>

<math display="block"> \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.</math>

Consider the problem of finding solutions of the form {{math|1=''f''(''r'', ''θ'', ''φ'') = ''R''(''r'') ''Y''(''θ'', ''φ'')}}. By [[Separation of variables#pde|separation of variables]], two differential equations result by imposing Laplace's equation:
<math display="block">\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^2Y}{\partial\varphi^2} = -\lambda.</math>
The second equation can be simplified under the assumption that {{math|''Y''}} has the form {{math|1=''Y''(''θ'', ''φ'') = Θ(''θ'') Φ(''φ'')}}. Applying separation of variables again to the second equation gives way to the pair of differential equations

<math display="block">\frac{1}{\Phi} \frac{d^2 \Phi}{d\varphi^2} = -m^2</math>
<math display="block">\lambda\sin^2\theta + \frac{\sin\theta}{\Theta} \frac{d}{d\theta} \left(\sin\theta \frac{d\Theta}{d\theta}\right) = m^2</math>

for some number {{math|''m''}}. A priori, {{math|''m''}} is a complex constant, but because {{math|Φ}} must be a [[Periodic function|periodic function]] whose period evenly divides {{math|2''π''}}, {{math|''m''}} is necessarily an integer and {{math|Φ}} is a linear combination of the complex exponentials {{math|''e''<sup>± ''imφ''</sup>}}. The solution function {{math|''Y''(''θ'', ''φ'')}} is regular at the poles of the sphere, where {{math|1=''θ'' = 0, ''π''}}. Imposing this regularity in the solution {{math|Θ}} of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter {{math|''λ''}} to be of the form {{math|1=''λ'' = ''ℓ'' (''ℓ'' + 1)}} for some non-negative integer with {{math|''ℓ'' ≥ {{!}}''m''{{!}}}}; this is also explained below in terms of the [[Physics:Angular momentum operator|orbital angular momentum]]. Furthermore, a change of variables {{math|1=''t'' = cos ''θ''}} transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial {{math|''P{{su|b=ℓ|p=m}}''(cos ''θ'')}} . Finally, the equation for {{math|''R''}} has solutions of the form {{math|1=''R''(''r'') = ''A r{{i sup|ℓ}}'' + ''B r''{{i sup|−''ℓ'' − 1}}}}; requiring the solution to be regular throughout {{math|'''R'''<sup>3</sup>}} forces {{math|1=''B'' = 0}}.<ref>Physical applications often take the solution that vanishes at infinity, making {{math|1=''A'' = 0}}. This does not affect the angular portion of the spherical harmonics.</ref>

Here the solution was assumed to have the special form {{math|1=''Y''(''θ'', ''φ'') = Θ(''θ'') Φ(''φ'')}}. For a given value of {{math|''ℓ''}}, there are {{math|2''ℓ'' + 1}} independent solutions of this form, one for each integer {{math|''m''}} with {{math|−''ℓ'' ≤ ''m'' ≤ ''ℓ''}}. These angular solutions <math>Y_{\ell}^m : S^2 \to \Complex</math> are a product of trigonometric functions, here represented as a [[Euler's formula|complex exponential]], and associated Legendre polynomials:

<math display="block"> Y_\ell^m (\theta, \varphi ) = N e^{i m \varphi } P_\ell^m (\cos{\theta} )</math>

which fulfill
<math display="block"> r^2\nabla^2 Y_\ell^m (\theta, \varphi ) = -\ell (\ell + 1 ) Y_\ell^m (\theta, \varphi ).</math>

Here <math>Y_{\ell}^m:S^2 \to \Complex</math> is called a '''spherical harmonic function of degree {{math|''ℓ''}} and order {{math|''m''}}''', <math>P_{\ell}^m:[-1,1]\to \R</math> is an associated Legendre polynomial, {{math|''N''}} is a normalization constant,<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Spherical Harmonic |url=https://mathworld.wolfram.com/ |access-date=2023-05-10 |website=mathworld.wolfram.com |language=en}}</ref> and {{math|''θ''}} and {{math|''φ''}} represent colatitude and longitude, respectively. In particular, the [[Colatitude|colatitude]] {{math|''θ''}}, or polar angle, ranges from {{math|''0''}} at the North Pole, to {{math|''π''/2}} at the Equator, to {{math|''π''}} at the South Pole, and the [[Earth:Longitude|longitude]] {{math|''φ''}}, or [[Azimuth|azimuth]], may assume all values with {{math|0 ≤ ''φ'' < 2''π''}}. For a fixed integer {{math|''ℓ''}}, every solution {{math|''Y''(''θ'', ''φ'')}}, <math>Y: S^2 \to \Complex</math>, of the eigenvalue problem
<math display="block"> r^2\nabla^2 Y = -\ell (\ell + 1 ) Y</math>
is a [[Linear combination|linear combination]] of <math>Y_\ell^m : S^2 \to \Complex</math>. In fact, for any such solution, {{math|''r<sup>ℓ</sup> Y''(''θ'', ''φ'')}} is the expression in spherical coordinates of a [[Homogeneous polynomial|homogeneous polynomial]] <math>\R^3 \to \Complex</math> that is harmonic (see below), and so counting dimensions shows that there are {{math|2''ℓ'' + 1}} linearly independent such polynomials.

The general solution <math>f:\R^3 \to \Complex</math> to [[Laplace's equation]] <math>\Delta f = 0</math> in a ball centered at the origin is a [[Linear combination|linear combination]] of the spherical harmonic functions multiplied by the appropriate scale factor {{math|''r<sup>ℓ</sup>''}},

<math display="block"> f(r, \theta, \varphi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m r^\ell Y_\ell^m (\theta, \varphi ), </math>

where the <math>f_{\ell}^m \in \Complex</math> are constants and the factors {{math|''r<sup>ℓ</sup> Y<sub>ℓ</sub><sup>m</sup>''}} are known as (''regular'') [[Physics:Solid harmonics|solid harmonics]] <math>\R^3 \to \Complex</math>. Such an expansion is valid in the [[Ball (mathematics)|ball]]

<math display="block"> r < R = \frac{1}{\limsup_{\ell\to\infty} |f_\ell^m|^{{1}/{\ell}}}.</math>

For <math> r > R</math>, the solid harmonics with negative powers of <math>r</math> (the ''irregular'' [[Physics:Solid harmonics|solid harmonics]] <math>\R^3 \setminus \{ \mathbf{0} \} \to \Complex</math>) are chosen instead. In that case, one needs to expand the solution of known regions in [[Laurent series]] (about <math>r=\infty</math>), instead of the [[Taylor series]] (about <math>r = 0</math>) used above, to match the terms and find series expansion coefficients <math>f^m_\ell \in \Complex</math>.

===Orbital angular momentum===
In quantum mechanics, Laplace's spherical harmonics are understood in terms of the [[Physics:Angular momentum operator|orbital angular momentum]]<ref>{{harvnb|Edmonds|1957|loc=§2.5}}</ref>
<math display="block">\mathbf{L} = -i\hbar (\mathbf{x}\times \mathbf{\nabla}) = L_x\mathbf{i} + L_y\mathbf{j}+L_z\mathbf{k}.</math>
The {{math|''ħ''}} is conventional in quantum mechanics; it is convenient to work in units in which {{math|1=''ħ'' = 1}}. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum
<math display="block">\begin{align}
\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{align}</math>
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:
<math display="block">\begin{align}
L_z &= -i\left(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}\right)\\
&=-i\frac{\partial}{\partial\varphi}.
\end{align}</math>

These operators commute, and are [[Densely defined operator|densely defined]] [[Self-adjoint operator|self-adjoint operator]]s on the [[Lp space#Weighted Lp spaces|weighted]] [[Hilbert space]] of functions ''f'' square-integrable with respect to the [[Normal distribution|normal distribution]] as the weight function on '''R'''<sup>3</sup>:
<math display="block">\frac{1}{(2\pi)^{3/2}}\int_{\R^3} |f(x)|^2 e^{-|x|^2/2}\,dx < \infty.</math>
Furthermore, '''L'''<sup>2</sup> is a positive operator.

If {{math|''Y''}} is a joint eigenfunction of {{math|'''L'''<sup>2</sup>}} and {{math|''L''<sub>''z''</sub>}}, then by definition
<math display="block">\begin{align}
\mathbf{L}^2Y &= \lambda Y\\
L_zY &= mY
\end{align}</math>
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 2''π''. Furthermore, since
<math display="block">\mathbf{L}^2 = L_x^2 + L_y^2 + L_z^2</math>
and each of ''L''<sub>''x''</sub>, ''L''<sub>''y''</sub>, ''L''<sub>''z''</sub> are self-adjoint, it follows that {{math|''λ'' ≥ ''m''<sup>2</sup>}}.

Denote this joint eigenspace by {{math|''E''<sub>''λ'',''m''</sub>}}, and define the raising and lowering operators by
<math display="block">\begin{align}
L_+ &= L_x + iL_y\\
L_- &= L_x - iL_y
\end{align}</math>
Then {{math|''L''<sub>+</sub>}} and {{math|''L''<sub>−</sub>}} commute with {{math|'''L'''<sup>2</sup>}}, and the Lie algebra generated by {{math|''L''<sub>+</sub>}}, {{math|''L''<sub>−</sub>}}, {{math|''L''<sub>''z''</sub>}} is the [[Special linear Lie algebra|special linear Lie algebra]] of order 2, <math>\mathfrak{sl}_2(\Complex)</math>, with commutation relations
<math display="block">[L_z,L_+] = L_+,\quad [L_z,L_-] = -L_-, \quad [L_+,L_-] = 2L_z.</math>
Thus {{math|''L''<sub>+</sub> : ''E''<sub>''λ'',''m''</sub> → ''E''<sub>''λ'',''m''+1</sub>}} (it is a "raising operator") and {{math|''L''<sub>−</sub> : ''E''<sub>''λ'',''m''</sub> → ''E''<sub>''λ'',''m''−1</sub>}} (it is a "lowering operator"). In particular, {{math|1=''L''{{su|b=+|p=''k''}} : ''E''<sub>''λ'',''m''</sub> → ''E''<sub>''λ'',''m''+''k''</sub>}} must be zero for ''k'' sufficiently large, because the inequality {{mvar|''λ'' ≥ ''m''<sup>2</sup>}} must hold in each of the nontrivial joint eigenspaces. Let {{mvar|''Y'' ∈ ''E''<sub>''λ'',''m''</sub>}} be a nonzero joint eigenfunction, and let {{mvar|k}} be the least integer such that
<math display="block">L_+^kY = 0.</math>
Then, since
<math display="block">L_-L_+ = \mathbf{L}^2 - L_z^2 - L_z</math>
it follows that
<math display="block">0 = L_-L_+^k Y = (\lambda - (m+k)^2-(m+k))Y.</math>
Thus {{math|1=''λ'' = ''ℓ''(''ℓ'' + 1)}} for the positive integer {{math|1=''ℓ'' = ''m'' + ''k''}}.

The foregoing has been all worked out in the spherical coordinate representation, <math>\langle \theta, \varphi| l m\rangle = Y_l^m (\theta, \varphi)</math> but may be expressed more abstractly in the complete, orthonormal [[Physics:Angular momentum operator#Orbital angular momentum in spherical coordinates|spherical ket basis]].

==Harmonic polynomial representation==

The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions <math>\R^3 \to \Complex</math>. Specifically, we say that a (complex-valued) polynomial function <math>p: \R^3 \to \Complex</math> is [[Homogeneous polynomial|''homogeneous'']] of degree <math>\ell</math> if
<math display="block">p(\lambda\mathbf x)=\lambda^\ell p(\mathbf x)</math>
for all real numbers <math>\lambda \in \R</math> and all <math>x \in \R^3</math>. We say that <math>p</math> is [[Harmonic function|''harmonic'']] if
<math display="block">\Delta p=0,</math>
where <math>\Delta</math> is the Laplacian. Then for each <math>\ell</math>, we define
<math display="block">\mathbf{A}_\ell = \left\{\text{harmonic polynomials } \R^3 \to \Complex \text{ that are homogeneous of degree } \ell \right\}.</math>

For example, when <math>\ell=1</math>, <math>\mathbf{A}_1</math> is just the 3-dimensional space of all linear functions <math>\R^3 \to \Complex</math>, since any such function is automatically harmonic. Meanwhile, when <math>\ell = 2</math>, we have a 5-dimensional space:
<math display="block">\mathbf{A}_2 = \operatorname{span}_{\Complex}(x_1 x_2,\, x_1 x_3,\, x_2 x_3,\, x_1^2-x_2^2,\, x_1^2-x_3^2).</math>

For any <math>\ell</math>, the space <math>\mathbf{H}_{\ell}</math> of spherical harmonics of degree <math>\ell</math> is just the space of restrictions to the sphere <math>S^2</math> of the elements of <math>\mathbf{A}_\ell</math>.<ref>{{harvnb|Hall|2013}} Section 17.6</ref> 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|harmonic function]]).

For example, for any <math>c \in \Complex</math> the formula
<math display="block">p(x_1, x_2, x_3) = c(x_1 + ix_2)^\ell</math>
defines a homogeneous polynomial of degree <math>\ell</math> with domain and codomain <math>\R^3 \to \Complex</math>, which happens to be independent of <math>x_3</math>. This polynomial is easily seen to be harmonic. If we write <math>p</math> in spherical coordinates <math>(r,\theta,\varphi)</math> and then restrict to <math>r = 1</math>, we obtain
<math display="block">p(\theta,\varphi) = c \sin(\theta)^\ell (\cos(\varphi) + i \sin(\varphi))^\ell,</math>
which can be rewritten as
<math display="block">p(\theta,\varphi) = c\left(\sqrt{1-\cos^2(\theta)}\right)^\ell e^{i\ell\varphi}.</math>
After using the formula for the [[Physics:Associated Legendre polynomials|associated Legendre polynomial]] <math>P^\ell_\ell</math>, we may recognize this as the formula for the spherical harmonic <math>Y^\ell_\ell(\theta, \varphi).</math><ref>{{harvnb|Hall|2013}} Lemma 17.16</ref> (See the section below on special cases of the spherical harmonics.)

==Conventions==

===Orthogonality and normalization===
{{Disputed section|Condon-Shortley phase|date = December 2017}}

Several different normalizations are in common use for the Laplace spherical harmonic functions <math>S^2 \to \Complex</math>. Throughout the section, we use the standard convention that for <math>m>0</math> (see [[Physics:Associated Legendre polynomials|associated Legendre polynomials]])
<math display="block">P_\ell ^{-m} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} P_\ell ^{m}</math>
which is the natural normalization given by Rodrigues' formula.
[[File:Plot of the spherical harmonic Y l^m(theta,phi) with n=2 and m=1 and phi=pi in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the spherical harmonic Y l^m(theta,phi) with n=2 and m=1 and phi=pi in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the spherical harmonic <math>Y_\ell^m(\theta,\varphi)</math> with <math>\ell=2</math> and <math>m=1</math> and <math>\varphi=\pi</math> in the complex plane from <math>-2-2i</math> to <math>2+2i</math> with colors created with Mathematica 13.1 function ComplexPlot3D]]
In [[Physics:Acoustics|acoustics]],<ref>{{Cite book|title=Fourier acoustics : sound radiation and nearfield acoustical holography |last=Williams |first=Earl G. |date=1999|publisher=Academic Press|isbn=0080506909|location=San Diego, Calif.|oclc=181010993}}</ref> the Laplace spherical harmonics are generally defined as (this is the convention used in this article)
<math display="block"> Y_\ell^m( \theta , \varphi ) = \sqrt{\frac{(2\ell+1)}{4\pi} \frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\theta} ) \, e^{i m \varphi } </math>
while in [[Physics:Quantum mechanics|quantum mechanics]]:<ref>{{cite book|last=Messiah|first=Albert|title=Quantum mechanics : two volumes bound as one| year=1999|publisher=Dover|location=Mineola, NY|isbn=9780486409245|edition=Two vol. bound as one, unabridged reprint}}</ref><ref>{{cite book|author1=Claude Cohen-Tannoudji | author2=Bernard Diu | author3=Franck Laloë |translator=Susan Reid Hemley |display-translators=etal | title=Quantum mechanics | year=1996 | publisher=Wiley | location=Wiley-Interscience | isbn=9780471569527}}</ref>
<math display="block"> Y_\ell^m( \theta , \varphi ) = (-1)^m \sqrt{\frac{(2\ell+1)}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} \, P_{\ell}^m ( \cos{\theta} ) \, e^{i m \varphi } </math>

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

In both definitions, the spherical harmonics are orthonormal
<math display="block">\int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^m \, Y_{\ell'}^{m'}{}^* \, d\Omega=\delta_{\ell\ell'}\, \delta_{mm'},</math>
where {{math|''δ''<sub>''ij''</sub>}} is the [[Kronecker delta]] and {{math|1=''d''Ω = sin(''θ'') ''dφ'' ''dθ''}}. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.,
<math display="block">\int{|Y_\ell^m|^2 d\Omega} = 1.</math>

The disciplines of [[Geodesy|geodesy]]<ref name="Blakely 1995 113">{{cite book | last = Blakely | first = Richard | title = Potential theory in gravity and magnetic applications | url = https://archive.org/details/potentialtheoryg00blak_461 | url-access = limited | publisher = Cambridge University Press | location = Cambridge England New York | year = 1995 | isbn = 978-0521415088 | page=[https://archive.org/details/potentialtheoryg00blak_461/page/n132 113] }}</ref> and spectral analysis use

<math display="block"> Y_\ell^m( \theta , \varphi ) = \sqrt{{(2\ell+1) }\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\theta} )\, e^{i m \varphi } </math>

which possess unit power

<math display="block">\frac{1}{4 \pi} \int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^m \, Y_{\ell'}^{m'}{}^* d\Omega=\delta_{\ell\ell'}\, \delta_{mm'}.</math>

The [[Physics:Magnetism|magnetics]]<ref name="Blakely 1995 113"/> community, in contrast, uses Schmidt semi-normalized harmonics

<math display="block"> Y_\ell^m( \theta , \varphi ) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\theta} ) \, e^{i m \varphi }</math>

which have the normalization

<math display="block"> \int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^m \, Y_{\ell'}^{m'}{}^*d\Omega = \frac{4 \pi}{(2 \ell + 1)} \delta_{\ell\ell'}\, \delta_{mm'}.</math>

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

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

<math display="block">Y_\ell^{m}{}^* (\theta, \varphi) = (-1)^m Y_\ell^{-m} (\theta, \varphi),</math>

where the superscript {{math|*}} denotes complex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with the [[Wigner D-matrix#Relation to spherical harmonics and Legendre polynomials|Wigner D-matrix]].

===Condon–Shortley phase===
One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of <math>(-1)^m</math>, commonly referred to as the [[Biography:Edward Condon|Condon]]–Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include this [[Phase factor|phase factor]] in the definition of the [[Physics:Associated Legendre polynomials|associated 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 of [[Physics:Ladder operator|raising and lowering operators]]. The geodesy<ref>Heiskanen and Moritz, Physical Geodesy, 1967, eq. 1-62</ref> 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.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Condon-Shortley Phase |url=https://mathworld.wolfram.com/Condon-ShortleyPhase.html |access-date=2022-11-02 |website=mathworld.wolfram.com |language=en}}</ref>

===Real form===
A real basis of spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> can be defined in terms of their complex analogues <math>Y_{\ell}^m: S^2 \to \Complex</math> by setting
<math display="block">\begin{align}
Y_{\ell m} &= \begin{cases}
\dfrac{i}{\sqrt{2}} \left(Y_\ell^{m} - (-1)^m\, Y_\ell^{-m}\right) & \text{if}\ m < 0\\
Y_\ell^0 & \text{if}\ m=0\\
\dfrac{1}{\sqrt{2}} \left(Y_\ell^{-m} + (-1)^m\, Y_\ell^{m}\right) & \text{if}\ m > 0.
\end{cases}\\
&= \begin{cases}
\dfrac{i}{\sqrt{2}} \left(Y_\ell^{-|m|} - (-1)^{m}\, Y_\ell^{|m|}\right) & \text{if}\ m < 0\\
Y_\ell^0 & \text{if}\ m=0\\
\dfrac{1}{\sqrt{2}} \left(Y_\ell^{-|m|} + (-1)^{m}\, Y_\ell^{|m|}\right) & \text{if}\ m>0.
\end{cases}\\
&= \begin{cases}
\sqrt{2} \, (-1)^m \, \Im [{Y_\ell^{|m|}}] & \text{if}\ m<0\\
Y_\ell^0 & \text{if}\ m=0\\
\sqrt{2} \, (-1)^m \, \Re [{Y_\ell^m}] & \text{if}\ m>0.
\end{cases}
\end{align}
</math>
The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics <math>Y_{\ell}^m : S^2 \to \Complex</math> in terms of the real spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> are
<math display="block">
Y_{\ell}^{m} = \begin{cases}
\dfrac{1}{\sqrt{2}} \left(Y_{\ell |m|} - i Y_{\ell,-|m|}\right) & \text{if}\ m<0 \\[4pt]
Y_{\ell 0} &\text{if}\ m=0 \\[4pt]
\dfrac{(-1)^m}{ \sqrt{2}} \left(Y_{\ell |m|} + i Y_{\ell,-|m|}\right) & \text{if}\ m>0.
\end{cases}
</math>

The real spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> are sometimes known as ''tesseral spherical harmonics''.<ref>{{harvnb|Whittaker|Watson|1927|p=392}}.</ref> These functions have the same orthonormality properties as the complex ones <math>Y_{\ell}^m : S^2 \to \Complex</math> above. The real spherical harmonics <math>Y_{\ell m}</math> with {{math|''m'' > 0}} are said to be of cosine type, and those with {{math|''m'' < 0}} of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as
<math display="block">
Y_{\ell m} = \begin{cases}
 \left(-1\right)^m\sqrt{2} \sqrt{\dfrac{2\ell+1}{4\pi}\dfrac{(\ell-|m|)!}{(\ell+|m|)!}} \;
 P_\ell^{|m|}(\cos \theta) \ \sin( |m|\varphi )
 &\text{if } m<0
\\[4pt]
 \sqrt{\dfrac{ 2\ell+1}{4\pi}} \ P_\ell^m(\cos \theta)
 & \text{if } m=0
\\[4pt]
 \left(-1\right)^m\sqrt{2} \sqrt{\dfrac{2\ell+1}{4\pi}\dfrac{(\ell-m)!}{(\ell+m)!}} \;
 P_\ell^m(\cos \theta) \ \cos( m\varphi )
 & \text{if } m>0 \,.
\end{cases}
</math>

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

See [[Table of spherical harmonics#Real spherical harmonics|here]] for a list of real spherical harmonics up to and including <math>\ell = 4</math>, which can be seen to be consistent with the output of the equations above.

==== Use in quantum chemistry ====

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, it is important to note that the real functions span the same space as the complex ones would.

For example, as can be seen from the [[Table of spherical harmonics#Spherical harmonics|table of spherical harmonics]], the usual {{math|''p''}} functions (<math>\ell = 1</math>) are complex and mix axis directions, but the [[Table of spherical harmonics#Real spherical harmonics|real versions]] are essentially just {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}.

== Spherical harmonics in Cartesian form ==

The complex spherical harmonics <math>Y_\ell^m</math> give rise to the [[Physics:Solid harmonics|solid harmonics]] by extending from <math>S^2</math> to all of <math>\R^3</math> as a [[Homogeneous function|homogeneous function]] of degree <math>\ell</math>, i.e. setting
<math display="block">R_\ell^m(v) := \|v\|^\ell Y_\ell^m\left(\frac{v}{\|v\|}\right)</math>
It turns out that <math>R_\ell^m</math> is basis of the space of harmonic and [[Homogeneous polynomial|homogeneous polynomial]]s of degree <math>\ell</math>. More specifically, it is the (unique up to normalization) Gelfand-Tsetlin-basis of this representation of the rotational group <math>SO(3)</math> and an [[Physics:Solid harmonics#Complex form|explicit formula]] for <math>R_\ell^m</math> in cartesian coordinates can be derived from that fact.

===The Herglotz generating function===

If the quantum mechanical convention is adopted for the <math>Y_{\ell}^m: S^2 \to \Complex</math>, then
<math display="block">
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).
</math>
Here, <math>\mathbf r</math> is the vector with components <math>(x, y, z) \in \R^3</math>, <math>r = |\mathbf{r}|</math>, and
<math display="block">
{\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).
</math>
<math>\mathbf a </math>     is a vector with complex coordinates:

<math>\mathbf a =[\frac{1}{2}(\frac{1}{\lambda}-\lambda),-\frac{i}{2 }(\frac{1}{\lambda} +\lambda),1       ] .</math>

The essential property of <math>\mathbf a</math> is that it is null:
<math display="block">\mathbf a \cdot \mathbf a = 0.</math>

It suffices to take <math>v</math> and <math>\lambda</math> as real parameters.
In naming this generating function after Herglotz, we follow {{harvnb|Courant|Hilbert|1962|loc= §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.<ref>See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012).</ref> An immediate benefit of this definition is that if the vector <math>\mathbf r</math> is replaced by the quantum mechanical spin vector operator <math>\mathbf J</math>, such that <math>\mathcal{Y}_{\ell}^m({\mathbf J})</math> is the operator analogue of the [[Physics:Solid harmonics|solid harmonic]] <math>r^{\ell}Y_{\ell}^m (\mathbf{r}/r)</math>,<ref>{{Citation| last1=Li|first1=Feifei| last2=Braun|first2=Carol| last3=Garg|first3=Anupam|title=The Weyl-Wigner-Moyal Formalism for Spin|journal=Europhysics Letters|year=2013|volume=102|issue=6| page=60006|doi=10.1209/0295-5075/102/60006|arxiv=1210.4075|bibcode=2013EL....10260006L|s2cid=119610178}}</ref> one obtains a generating function for a standardized set of spherical tensor operators, <math>\mathcal{Y}_{\ell}^m({\mathbf J})</math>:

<math display="block">
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}).
</math>

The parallelism of the two definitions ensures that the <math>\mathcal{Y}_{\ell}^m</math>'s transform under rotations (see below) in the same way as the <math>Y_{\ell}^m</math>'s, which in turn guarantees that they are spherical tensor operators, <math>T^{(k)}_q</math>, with <math>k = {\ell}</math> and <math>q = m</math>, 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.


===Separated Cartesian form===
The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of <math>z</math> and another of <math>x</math> and <math>y</math>, as follows (Condon–Shortley phase):
<math display="block">
r^\ell\,
\begin{pmatrix} Y_\ell^{m} \\ Y_\ell^{-m} \end{pmatrix}
=
\left[\frac{2\ell+1}{4\pi}\right]^{1/2} \bar{\Pi}^m_\ell(z)
\begin{pmatrix} \left(-1\right)^m (A_m + i B_m) \\ (A_m - i B_m) \end{pmatrix} ,
\qquad m > 0.
</math>
and for {{math|1=''m'' = 0}}:
<math display="block">r^\ell\,Y_\ell^{0} \equiv \sqrt{\frac{2\ell+1}{4\pi}} \bar{\Pi}^0_\ell .</math>
Here
<math display="block">A_m(x,y) = \sum_{p=0}^m \binom{m}{p} x^p y^{m-p} \cos \left((m-p) \frac{\pi}{2}\right),</math>
<math display="block">B_m(x,y) = \sum_{p=0}^m \binom{m}{p} x^p y^{m-p} \sin \left((m-p) \frac{\pi}{2}\right),</math>
and
<math display="block">
\bar{\Pi}^m_\ell(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}.
</math>
For <math>m = 0</math> this reduces to
<math display="block">
\bar{\Pi}^0_\ell(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}.
</math>

The factor <math>\bar{\Pi}_\ell^m(z)</math> is essentially the associated Legendre polynomial <math>P_\ell^m(\cos\theta)</math>, and the factors <math>(A_m \pm i B_m)</math> are essentially <math>e^{\pm i m\varphi}</math>.

====Examples====

Using the expressions for <math>\bar{\Pi}_\ell^m(z)</math>, <math>A_m(x,y)</math>, and <math>B_m(x,y)</math> listed explicitly above we obtain:
<math display="block">
 Y^1_3
= - \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)
</math>

<math display="block">
Y^{-2}_4 = \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^{-2 i \varphi}\right)
</math>
It may be verified that this agrees with the function listed [[Table of spherical harmonics#ℓ = 3|here]] and [[Table of spherical harmonics#ℓ = 4|here]].

====Real forms====

Using the equations above to form the real spherical harmonics, it is seen that for <math>m>0</math> only the <math>A_m</math> terms (cosines) are included, and for <math>m<0</math> only the <math>B_m</math> terms (sines) are included:

<math display="block">
r^\ell\,
\begin{pmatrix} Y_{\ell m} \\ Y_{\ell -m} \end{pmatrix}
=
\sqrt{\frac{2\ell+1}{2\pi}} \bar{\Pi}^m_\ell(z)
\begin{pmatrix} A_m \\ B_m \end{pmatrix} ,
\qquad m > 0.
</math>
and for ''m'' = 0:
<math display="block">
r^\ell\,Y_{\ell 0} \equiv \sqrt{\frac{2\ell+1}{4\pi}}
\bar{\Pi}^0_\ell .
</math>

==Special cases and values==
# When <math>m = 0</math>, the spherical harmonics <math>Y_{\ell}^m: S^2 \to \Complex</math> reduce to the ordinary [[Legendre polynomials]]: <math display="block">Y_{\ell}^0(\theta, \varphi) = \sqrt{\frac{2\ell+1}{4\pi}} P_{\ell}(\cos\theta).</math>
# When <math>m = \pm\ell</math>, <math display="block">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},</math> or more simply in Cartesian coordinates, <math display="block">r^{\ell} Y_{\ell}^{\pm\ell}({\mathbf r})
= \frac{(\mp 1)^{\ell}}{2^{\ell}\ell!} \sqrt{\frac{(2\ell+1)!}{4\pi}} (x \pm i y)^{\ell}.</math>
# At the north pole, where <math> \theta = 0</math>, and <math>\varphi</math> is undefined, all spherical harmonics except those with <math>m = 0</math> vanish: <math display="block"> Y_{\ell}^m(0,\varphi) = Y_{\ell}^m({\mathbf z}) = \sqrt{\frac{2\ell+1}{4\pi}} \delta_{m0}.</math>

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

=== Parity ===
{{Main|Physics:Parity}}
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 <math>P\Psi(\mathbf r) = \Psi(-\mathbf r)</math>. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with <math>\mathbf r</math> being a unit vector,
<math display="block"> Y_\ell^m(-\mathbf r) = (-1)^\ell Y_\ell^m(\mathbf r).</math>

In terms of the spherical angles, parity transforms a point with coordinates <math>\{\theta,\varphi\}</math> to <math>\{\pi-\theta,\pi+\varphi\}</math>. The statement of the parity of spherical harmonics is then
<math display="block">Y_\ell^m(\theta,\varphi) \to Y_\ell^m(\pi-\theta,\pi+\varphi) = (-1)^\ell Y_\ell^m(\theta,\varphi)</math>
(This can be seen as follows: The associated Legendre polynomials gives {{math|(−1)<sup>''ℓ''+''m''</sup>}} and from the exponential function we have {{math|(−1)<sup>''m''</sup>}}, giving together for the spherical harmonics a parity of {{math|(−1)<sup>''ℓ''</sup>}}.)

Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a [[Point reflection|point reflection]] to a spherical harmonic of degree {{mvar|ℓ}} changes the sign by a factor of {{math|(−1)<sup>''ℓ''</sup>}}.

===Rotations===

[[File:Rotation of octupole vector function.svg|thumb|500px|The rotation of a real spherical function with {{math|1=''m'' = 0}} and {{math|1=''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 <math>\mathcal R</math> about the origin that sends the unit vector <math>\mathbf r</math> to <math>\mathbf r'</math>. Under this operation, a spherical harmonic of degree <math>\ell</math> and order <math>m</math> transforms into a linear combination of spherical harmonics of the same degree. That is,
<math display="block">
Y_\ell^m({\mathbf r}') = \sum_{m' = -\ell}^\ell A_{mm'} Y_\ell^{m'}({\mathbf r}),
</math>
where <math>A_{mm'}</math> is a matrix of order <math>(2\ell + 1)</math> that depends on the rotation <math>\mathcal R</math>. However, this is not the standard way of expressing this property. In the standard way one writes,

<math display="block">Y_\ell^m({\mathbf r}') = \sum_{m' = -\ell}^\ell [D^{(\ell)}_{mm'}({\mathcal R})]^* Y_\ell^{m'}({\mathbf r}),</math>
where <math>D^{(\ell)}_{mm'}({\mathcal R})^*</math> is the complex conjugate of an element of the [[Wigner D-matrix]]. In particular when <math>\mathbf r'</math> is a <math>\phi_0</math> rotation of the azimuth we get the identity,

<math display="block">Y_\ell^m({\mathbf r}') = Y_\ell^{m}({\mathbf r}) e^{i m \phi_0}.</math>

The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The <math>Y_\ell^m</math>'s of degree <math>\ell</math> provide a basis set of functions for the irreducible representation of the group SO(3) of dimension <math>(2\ell + 1)</math>. 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.

==Spherical harmonics expansion==
The Laplace spherical harmonics <math>Y_{\ell}^m:S^2 \to \Complex</math> form a complete set of orthonormal functions and thus form an [[Orthonormal basis|orthonormal basis]] of the [[Hilbert space]] of [[Square-integrable function|square-integrable function]]s <math>L^2_{\Complex}(S^2)</math>. On the unit sphere <math>S^2</math>, any square-integrable function <math>f:S^2 \to \Complex</math> can thus be expanded as a linear combination of these:

<math display="block">f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi).</math>

This expansion holds in the sense of mean-square convergence — convergence in [[Lp space|L<sup>2</sup>]] of the sphere — which is to say that

<math display="block">\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.</math>

The expansion coefficients are the analogs of [[Fourier series|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:

<math display="block">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).</math>

If the coefficients decay in ''ℓ'' sufficiently rapidly — for instance, [[Exponential decay|exponentially]] — then the series also [[Uniform convergence|converges uniformly]] to ''f''.

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

<math display="block"> f(\theta, \varphi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_{\ell m} \, Y_{\ell m}(\theta, \varphi). </math>

The convergence of the series holds again in the same sense, namely the real spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> form a complete set of orthonormal functions and thus form an [[Orthonormal basis|orthonormal basis]] of the [[Hilbert space]] of [[Square-integrable function|square-integrable function]]s <math>L^2_{\R}(S^2)</math>. The benefit of the expansion in terms of the real harmonic functions <math>Y_{\ell m}</math> is that for real functions <math>f:S^2 \to \R</math> the expansion coefficients <math>f_{\ell m}</math> are guaranteed to be real, whereas their coefficients <math>f_{\ell}^m</math> in their expansion in terms of the <math>Y_{\ell}^m</math> (considering them as functions <math>f: S^2 \to \Complex \supset \R</math>) do not have that property.

==Spectrum analysis==

===Power spectrum in signal processing===
The total power of a function ''f'' is defined in the [[Signal processing|signal processing]] literature as the integral of the function squared, divided by the area of its domain. Using the [[Orthonormality|orthonormality]] 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 of [[Parseval's theorem]] (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics):

<math display="block">\frac{1}{4 \, \pi} \int_\Omega |f(\Omega)|^2\, d\Omega = \sum_{\ell=0}^\infty S_{f\!f}(\ell),</math>
where
<math display="block">S_{f\!f}(\ell) = \frac{1}{2\ell+1}\sum_{m=-\ell}^\ell |f_{\ell m}|^2 </math>

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
<math display="block">\frac{1}{4 \, \pi} \int_\Omega f(\Omega) \, g^\ast(\Omega) \, d\Omega = \sum_{\ell=0}^\infty S_{fg}(\ell),</math>
where
<math display="block">S_{fg}(\ell) = \frac{1}{2\ell+1}\sum_{m=-\ell}^\ell f_{\ell m} g^\ast_{\ell m} </math>

is defined as the cross-power spectrum. If the functions {{math|''f''}} and {{math|''g''}} have a zero mean (i.e., the spectral coefficients {{math|''f''<sub>00</sub>}} and {{math|''g''<sub>00</sub>}} are zero), then {{math|''S''<sub>''ff''</sub>(''ℓ'')}} and {{math|''S''<sub>''fg''</sub>(''ℓ'')}} represent the contributions to the function's variance and covariance for degree {{mvar|ℓ}}, respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form

<math display="block">S_{f\!f}(\ell) = C \, \ell^{\beta}.</math>

When {{math|1=''β'' = 0}}, the spectrum is "white" as each degree possesses equal power. When {{math|''β'' < 0}}, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when {{math|''β'' > 0}}, the spectrum is termed "blue". The condition on the order of growth of {{math|''S''<sub>''ff''</sub>(''ℓ'')}} is related to the order of differentiability of {{math|''f''}} in the next section.

===Differentiability properties===
One can also understand the [[Derivative|differentiability properties]] of the original function {{math|''f''}} in terms of the asymptotics of {{math|''S''<sub>''ff''</sub>(''ℓ'')}}. In particular, if {{math|''S''<sub>''ff''</sub>(''ℓ'')}} decays faster than any [[Rational function|rational function]] of {{mvar|ℓ}} as {{math|''ℓ'' → ∞}}, then {{math|''f''}} is infinitely differentiable. If, furthermore, {{math|''S''<sub>''ff''</sub>(''ℓ'')}} decays exponentially, then {{math|''f''}} is actually real analytic on the sphere.

The general technique is to use the theory of [[Sobolev space]]s. Statements relating the growth of the {{math|''S''<sub>''ff''</sub>(''ℓ'')}} to differentiability are then similar to analogous results on the growth of the coefficients of [[Fourier series]]. Specifically, if
<math display="block">\sum_{\ell=0}^\infty (1+\ell^2)^s S_{ff}(\ell) < \infty,</math>
then {{math|''f''}} is in the Sobolev space {{math|''H''<sup>''s''</sup>(''S''<sup>2</sup>)}}. In particular, the Sobolev embedding theorem implies that {{math|''f''}} is infinitely differentiable provided that
<math display="block">S_{ff}(\ell) = O(\ell^{-s})\quad\rm{as\ }\ell\to\infty</math>
for all {{math|''s''}}.

==Algebraic properties==

=== Addition theorem ===
A mathematical result of considerable interest and use is called the ''addition theorem'' for spherical harmonics. Given two vectors {{math|'''r'''}} and {{math|'''r′'''}}, with spherical coordinates <math>(r,\theta,\varphi)</math> and <math>(r ', \theta ', \varphi ')</math>, respectively, the angle <math>\gamma</math> between them is given by the relation
<math display="block">\cos\gamma = \cos\theta'\cos\theta + \sin\theta\sin\theta' \cos(\varphi-\varphi')</math>
in which the role of the trigonometric functions appearing on the right-hand side is played by the spherical harmonics and that of the left-hand side is played by the Legendre polynomials.

The ''addition theorem'' states<ref>{{cite book|last1=Edmonds|first1=A. R.|title=Angular Momentum In Quantum Mechanics| year=1996 | url=https://archive.org/details/angularmomentumq00edmo|url-access=limited|publisher=Princeton University Press|page=[https://archive.org/details/angularmomentumq00edmo/page/n68 63]}}</ref>
{{NumBlk||<math display="block">
P_\ell( \mathbf{x}\cdot\mathbf{y} ) = \frac{4\pi}{2\ell+1}\sum_{m=-\ell}^\ell Y_{\ell m}(\mathbf{y}) \, Y_{\ell m}^*(\mathbf{x})
\quad \forall \, \ell \in \N_0 \;
\forall\, \mathbf{x}, \mathbf{y} \in \R^3 \colon \; \| \mathbf{x} \|_2 = \| \mathbf{y} \|_2 = 1 \,,
</math>|{{EquationRef|1}}}}
where {{math|''P''<sub>''ℓ''</sub>}} is the Legendre polynomial of degree {{mvar|ℓ}}. This expression is valid for both real and complex harmonics.<ref>This is valid for any orthonormal basis of spherical harmonics of degree {{mvar|ℓ}}. For unit power harmonics it is necessary to remove the factor of {{math|4''π''}}.</ref> The result can be proven analytically, using the properties of the [[Poisson kernel]] in the unit ball, or geometrically by applying a rotation to the vector '''y''' so that it points along the ''z''-axis, and then directly calculating the right-hand side.<ref>{{harvnb|Whittaker|Watson|1927|p=395}}</ref>

In particular, when {{math|1='''x''' = '''y'''}}, this gives Unsöld's theorem<ref>{{harvnb|Unsöld|1927}}</ref>
<math display="block">\sum_{m=-\ell}^\ell Y_{\ell m}^*(\mathbf{x}) \, Y_{\ell m}(\mathbf{x}) = \frac{2\ell + 1}{4\pi}</math>
which generalizes the identity {{math|1=cos<sup>2</sup>''θ'' + sin<sup>2</sup>''θ'' = 1}} to two dimensions.

In the expansion ({{EquationNote|1}}), the left-hand side <math>P_{\ell} (\mathbf{x} \cdot \mathbf{y})</math> is a constant multiple of the degree {{mvar|ℓ}} zonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. Let {{math|''Y''<sub>''j''</sub>}} be an arbitrary orthonormal basis of the space {{math|'''H'''<sub>''ℓ''</sub>}} of degree {{mvar|ℓ}} spherical harmonics on the {{mvar|n}}-sphere. Then <math>Z^{(\ell)}_{\mathbf{x}}</math>, the degree {{mvar|ℓ}} zonal harmonic corresponding to the unit vector {{mvar|x}}, decomposes as<ref>{{harvnb|Stein|Weiss|1971|loc=§IV.2}}</ref>
{{NumBlk||<math display="block">Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}}) = \sum_{j=1}^{\dim(\mathbf{H}_\ell)}\overline{Y_j({\mathbf{x}})}\,Y_j({\mathbf{y}})</math>|{{EquationRef|2}}}}

Furthermore, the zonal harmonic <math>Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}})</math> is given as a constant multiple of the appropriate Gegenbauer polynomial:
{{NumBlk||<math display="block">Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}}) = C_\ell^{((n-2)/2)}({\mathbf{x}}\cdot {\mathbf{y}})</math>|{{EquationRef|3}}}}
Combining ({{EquationNote|2}}) and ({{EquationNote|3}}) gives ({{EquationNote|1}}) in dimension {{math|1=''n'' = 2}} when {{math|'''x'''}} and {{math|'''y'''}} are represented in spherical coordinates. Finally, evaluating at {{math|1='''x''' = '''y'''}} gives the functional identity
<math display="block">\frac{\dim \mathbf{H}_\ell}{\omega_{n-1}} = \sum_{j=1}^{\dim(\mathbf{H}_\ell)}|Y_j({\mathbf{x}})|^2</math>
where {{math|''ω''<sub>''n''−1</sub>}} is the volume of the (''n''−1)-sphere.

=== Contraction rule ===
Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics<ref>{{cite book | last1=Brink|first1=D. M.|last2=Satchler|first2=G. R.|title=Angular Momentum|publisher=Oxford University Press|page=146}}</ref>
<math display="block"> Y_{a,\alpha}\left(\theta,\varphi\right)Y_{b,\beta}\left(\theta,\varphi\right) = \sqrt{\frac{\left(2a+1\right) \left(2b+1\right)}{4\pi}}\sum_{c=0}^{\infty}\sum_{\gamma=-c}^{c}\left(-1\right)^{\gamma}\sqrt{2c+1}\begin{pmatrix}
a & b & c\\
\alpha & \beta & -\gamma
\end{pmatrix} \begin{pmatrix}
a & b & c\\
0 & 0 & 0
\end{pmatrix} Y_{c,\gamma}\left(\theta,\varphi\right). </math>
Many of the terms in this sum are trivially zero. The values of <math> c </math> and <math>\gamma</math> that result in non-zero terms in this sum are determined by the selection rules for the 3j-symbols.

=== Clebsch–Gordan coefficients ===
{{Main|Physics:Clebsch–Gordan coefficients}}
The Clebsch–Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the [[Physics:Slater integrals|Slater integrals]]. Abstractly, the Clebsch–Gordan coefficients express the [[Tensor product|tensor product]] of two [[Irreducible representation|irreducible representation]]s of the [[Rotation group SO(3)|rotation group]] as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.

==Visualization of the spherical harmonics==
[[Image:Spherical harmonics positive negative.svg|thumb|right|Schematic representation of <math>Y_{\ell m}</math> on the unit sphere and its nodal lines. <math>\Re [Y_{\ell m}]</math> is equal to 0 along {{mvar|m}} [[Great circle|great circle]]s passing through the poles, and along ''ℓ''−''m'' circles of equal latitude. The function changes sign each time it crosses one of these lines.]]
<!--
[[Image:Traces harmonique spherique.png|300px|thumb|right|The real spherical harmonic function <math>Y_{32}</math> shown in four cross sections.]]-->
[[Image:Spherical harmonics.png|300px|thumb|right|3D color plot of the spherical harmonics of degree {{math|1=''n'' = 5}}. Note that {{math|1=''n'' = ''ℓ''}}.]]

The Laplace spherical harmonics <math>Y_\ell^m</math> can be visualized by considering their "nodal lines", that is, the set of points on the sphere where <math>\Re [Y_\ell^m] = 0</math>, or alternatively where <math>\Im [Y_\ell^m] = 0</math>. Nodal lines of <math>Y_\ell^m</math> are composed of ''ℓ'' circles: there are {{math|{{abs|''m''}}}} circles along longitudes and ''ℓ''−|''m''| circles along latitudes. One can determine the number of nodal lines of each type by counting the number of zeros of <math>Y_\ell^m</math> in the <math>\theta</math> and <math>\varphi</math> directions respectively. Considering <math>Y_\ell^m</math> as a function of <math>\theta</math>, the real and imaginary components of the associated Legendre polynomials each possess ''ℓ''−|''m''| zeros, each giving rise to a nodal 'line of latitude'. On the other hand, considering <math>Y_\ell^m</math> as a function of <math>\varphi</math>, the trigonometric sin and cos functions possess 2|''m''| zeros, each of which gives rise to a nodal 'line of longitude'.

{{anchor|Zonal|Tesseral|Sectoral}}When the spherical harmonic order ''m'' is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as '''[[Zonal spherical harmonics|zonal]]'''. Such spherical harmonics are a special case of [[Zonal spherical function|zonal spherical function]]s. When {{math|1=''ℓ'' = {{abs|''m''}}}} (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as '''sectoral'''. For the other cases, the functions checker the sphere, and they are referred to as '''tesseral'''.

More general spherical harmonics of degree {{mvar|ℓ}} are not necessarily those of the Laplace basis <math>Y_\ell^m</math>, and their nodal sets can be of a fairly general kind.<ref>{{harvnb|Eremenko|Jakobson|Nadirashvili|2007}}</ref>

==List of spherical harmonics==
{{Main|Table of spherical harmonics}}

Analytic expressions for the first few orthonormalized Laplace spherical harmonics <math>Y_{\ell}^m : S^2 \to \Complex</math> that use the Condon–Shortley phase convention:
<math display="block">Y_{0}^{0}(\theta,\varphi) = \frac{1}{2}\sqrt{\frac 1 \pi}</math>

<math display="block">\begin{align}
Y_{1}^{-1}(\theta,\varphi) &= \frac{1}{2}\sqrt{\frac 3 {2\pi}} \, \sin\theta \, e^{-i\varphi} \\
Y_{1}^{0}(\theta,\varphi) &= \frac{1}{2}\sqrt{\frac 3 \pi}\, \cos\theta \\
Y_{1}^{1}(\theta,\varphi) &= \frac{-1}{2}\sqrt{\frac 3 {2\pi}}\, \sin\theta\, e^{i\varphi}
\end{align}</math>

<math display="block">\begin{align}
Y_{2}^{-2}(\theta,\varphi) &= \frac{1}{4}\sqrt{\frac{15}{2\pi}} \, \sin^{2}\theta \, e^{-2i\varphi} \\
Y_{2}^{-1}(\theta,\varphi) &= \frac{1}{2}\sqrt{\frac{15}{2\pi}}\, \sin\theta\, \cos\theta\, e^{-i\varphi} \\
Y_{2}^{0}(\theta,\varphi) &= \frac{1}{4} \sqrt{\frac{5}{\pi}}\, (3\cos^{2}\theta-1) \\
Y_{2}^{1}(\theta,\varphi) &= \frac{-1}{2}\sqrt{\frac{15}{2\pi}}\, \sin\theta\,\cos\theta\, e^{i\varphi} \\
Y_{2}^{2}(\theta,\varphi) &= \frac{1}{4}\sqrt{\frac{15}{2\pi}}\, \sin^{2}\theta \, e^{2i\varphi}
\end{align}</math>

==Higher dimensions==
The classical spherical harmonics are defined as complex-valued functions on the unit sphere <math>S^2</math> inside three-dimensional Euclidean space <math>\R^3</math>. Spherical harmonics can be generalized to higher-dimensional Euclidean space <math>\R^n</math> as follows, leading to functions <math>S^{n-1} \to \Complex</math>.<ref>{{harvnb |Solomentsev|2001}}; {{harvnb|Stein|Weiss|1971|loc=§Iv.2}}</ref> Let '''P'''<sub>''ℓ''</sub> denote the [[Vector space|space]] of complex-valued [[Homogeneous polynomial|homogeneous polynomial]]s of degree {{mvar|ℓ}} in {{mvar|n}} real variables, here considered as functions <math>\R^n \to \Complex</math>. That is, a polynomial {{math|''p''}} is in {{math|'''P'''<sub>''ℓ''</sub>}} provided that for any real <math>\lambda \in \R</math>, one has

<math display="block">p(\lambda \mathbf{x}) = \lambda^\ell p(\mathbf{x}).</math>

Let '''A'''<sub>''ℓ''</sub> denote the subspace of '''P'''<sub>''ℓ''</sub> consisting of all [[Harmonic polynomial|harmonic polynomial]]s:
<math display="block">\mathbf{A}_{\ell} := \{ p \in \mathbf{P}_{\ell} \,\mid\, \Delta p = 0 \} \,. </math>
These are the (regular) solid spherical harmonics. Let '''H'''<sub>''ℓ''</sub> denote the space of functions on the unit sphere
<math display="block">S^{n-1} := \{\mathbf{x}\in\R^n\,\mid\, \left|x\right|=1\}</math>
obtained by restriction from {{math|'''A'''<sub>''ℓ''</sub>}}
<math display="block">\mathbf{H}_{\ell} := \left\{ f: S^{n-1} \to \Complex \,\mid\, \text{ for some }p \in \mathbf{A}_{\ell},\, f(\mathbf{x}) = p(\mathbf{x}) \text{ for all }\mathbf{x} \in S^{n-1} \right\} .</math>

The following properties hold:
* The sum of the spaces {{math|'''H'''<sub>''ℓ''</sub>}} is [[Dense set|dense]] in the set <math>C(S^{n-1})</math> of continuous functions on <math>S^{n-1}</math> with respect to the [[Uniform convergence|uniform topology]], by the [[Stone–Weierstrass theorem]]. As a result, the sum of these spaces is also dense in the space {{math|''L''<sup>2</sup>(''S''<sup>''n''−1</sup>)}} of square-integrable functions on the sphere. Thus every square-integrable function on the sphere decomposes uniquely into a series of spherical harmonics, where the series converges in the {{math|''L''<sup>2</sup>}} sense.
* For all {{math|''f'' ∈ '''H'''<sub>''ℓ''</sub>}}, one has <math display="block">\Delta_{S^{n-1}}f = -\ell(\ell+n-2)f.</math> where {{math|Δ<sub>''S''<sup>''n''−1</sup></sub>}} is the [[Laplace–Beltrami operator]] on {{math|''S''<sup>''n''−1</sup>}}. This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian in {{mvar|n}} dimensions decomposes as <math display="block">\nabla^2
= r^{1-n}\frac{\partial}{\partial r}r^{n-1}\frac{\partial}{\partial r} + r^{-2}\Delta_{S^{n-1}}
= \frac{\partial^2}{\partial r^2} + \frac{n-1}{r}\frac{\partial}{\partial r} + r^{-2}\Delta_{S^{n-1}}</math>
* It follows from the Stokes theorem and the preceding property that the spaces {{math|'''H'''<sub>''ℓ''</sub>}} are orthogonal with respect to the inner product from {{math|''L''<sup>2</sup>(''S''<sup>''n''−1</sup>)}}. That is to say, <math display="block"> \int_{S^{n-1}} f\bar{g} \, \mathrm{d}\Omega = 0</math> for {{math|''f'' ∈ '''H'''<sub>''ℓ''</sub>}} and {{math|''g'' ∈ '''H'''<sub>''k''</sub>}} for {{math|''k'' ≠ ''ℓ''}}.
* Conversely, the spaces {{math|'''H'''<sub>''ℓ''</sub>}} are precisely the eigenspaces of {{math|Δ<sub>''S''<sup>''n''−1</sup></sub>}}. In particular, an application of the [[Spectral theorem#Compact self-adjoint operators|spectral theorem]] to the [[Riesz potential]] <math>\Delta_{S^{n-1}}^{-1}</math> gives another proof that the spaces {{math|'''H'''<sub>''ℓ''</sub>}} are pairwise orthogonal and complete in {{math|''L''<sup>2</sup>(''S''<sup>''n''−1</sup>)}}.
* Every homogeneous polynomial {{math|''p'' ∈ '''P'''<sub>''ℓ''</sub>}} can be uniquely written in the form<ref>Cf. Corollary 1.8 of {{Citation | title=Harmonic Polynomials and Dirichlet-Type Problems | last1=Axler | first1=Sheldon | last2=Ramey | first2=Wade | year=1995}}</ref> <math display="block">p(x) = p_\ell(x) + |x|^2p_{\ell-2} + \cdots + \begin{cases}
|x|^\ell p_0 & \ell \rm{\ even}\\
|x|^{\ell-1} p_1(x) & \ell\rm{\ odd}
\end{cases}</math> where {{math|''p''<sub>''j''</sub> ∈ '''A'''<sub>''j''</sub>}}. In particular, <math display="block">\dim \mathbf{H}_\ell = \binom{n+\ell-1}{n-1}-\binom{n+\ell-3}{n-1}=\binom{n+\ell-2}{n-2}+\binom{n+\ell-3}{n-2}.</math>

An orthogonal basis of spherical harmonics in higher dimensions can be constructed [[Mathematical induction|inductively]] by the method of [[Separation of variables|separation of variables]], by solving the Sturm-Liouville problem for the spherical Laplacian
<math display="block">\Delta_{S^{n-1}} = \sin^{2-n}\varphi\frac{\partial}{\partial\varphi}\sin^{n-2}\varphi\frac{\partial}{\partial\varphi} + \sin^{-2}\varphi \Delta_{S^{n-2}}</math>
where ''φ'' is the axial coordinate in a spherical coordinate system on ''S''<sup>''n''−1</sup>. The end result of such a procedure is<ref>{{cite journal|last=Higuchi|first=Atsushi|title=Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1)|journal=Journal of Mathematical Physics | year=1987 | volume=28 | issue=7 | pages=1553–1566 | doi=10.1063/1.527513 |url=https://aip.scitation.org/doi/10.1063/1.527513|bibcode = 1987JMP....28.1553H }}</ref>
<math display="block">Y_{\ell_1, \dots \ell_{n-1}} (\theta_1, \dots \theta_{n-1}) = \frac{1}{\sqrt{2\pi}} e^{i \ell_1 \theta_1} \prod_{j = 2}^{n-1} {}_j \bar{P}^{\ell_{j-1}}_{\ell_j} (\theta_j)</math>
where the indices satisfy {{math|{{abs|''ℓ''<sub>1</sub>}} ≤ ''ℓ''<sub>2</sub> ≤ ⋯ ≤ ''ℓ''<sub>''n''−1</sub>}} and the eigenvalue is {{math|−''ℓ''<sub>''n''−1</sub>(''ℓ''<sub>''n''−1</sub> + ''n''−2)}}. The functions in the product are defined in terms of the [[Legendre function]]
<math display="block">{}_j \bar{P}^\ell_{L} (\theta) = \sqrt{\frac{2L+j-1}{2} \frac{(L+\ell+j-2)!}{(L-\ell)!}} \sin^{\frac{2-j}{2}} (\theta) P^{-\left(\ell + \frac{j-2}{2}\right)}_{L+\frac{j-2}{2}} (\cos \theta) \,.</math>

==Connection with representation theory==
The space {{math|'''H'''<sub>''ℓ''</sub>}} of spherical harmonics of degree {{mvar|ℓ}} is a [[Group representation|representation]] of the symmetry [[Group (mathematics)|group]] of rotations around a point ([[SO(3)]]) and its double-cover [[Special unitary group|SU(2)]]. Indeed, rotations act on the two-dimensional [[Sphere|sphere]], and thus also on {{math|'''H'''<sub>''ℓ''</sub>}} by function composition
<math display="block"> \psi \mapsto \psi\circ\rho^{-1}</math>
for {{mvar|ψ}} a spherical harmonic and {{mvar|ρ}} a rotation. The representation {{math|'''H'''<sub>''ℓ''</sub>}} is an [[Irreducible representation|irreducible representation]] of SO(3).<ref>{{harvnb|Hall|2013}} Corollary 17.17</ref>

The elements of {{math|'''H'''<sub>''ℓ''</sub>}} arise as the restrictions to the sphere of elements of {{math|'''A'''<sub>''ℓ''</sub>}}: harmonic polynomials homogeneous of degree {{mvar|ℓ}} on three-dimensional Euclidean space {{math|'''R'''<sup>3</sup>}}. By [[Polarization of an algebraic form|polarization]] of {{math|''ψ'' ∈ '''A'''<sub>''ℓ''</sub>}}, there are coefficients <math>\psi_{i_1\dots i_\ell}</math> symmetric on the indices, uniquely determined by the requirement
<math display="block">\psi(x_1,\dots,x_n) = \sum_{i_1\dots i_\ell}\psi_{i_1\dots i_\ell}x_{i_1}\cdots x_{i_\ell}.</math>
The condition that {{mvar|ψ}} be harmonic is equivalent to the assertion that the [[Tensor|tensor]] <math>\psi_{i_1\dots i_\ell}</math> must be [[Tensor contraction|trace]] free on every pair of indices. Thus as an irreducible representation of {{math|1=SO(3)}}, {{math|'''H'''<sub>''ℓ''</sub>}} is isomorphic to the space of traceless [[Symmetric tensor|symmetric tensor]]s of degree {{mvar|ℓ}}.

More generally, the analogous statements hold in higher dimensions: the space {{math|'''H'''<sub>''ℓ''</sub>}} of spherical harmonics on the [[N-sphere|{{mvar|n}}-sphere]] is the irreducible representation of {{math|SO(''n''+1)}} corresponding to the traceless symmetric {{mvar|ℓ}}-tensors. However, whereas every irreducible tensor representation of {{math|SO(2)}} and {{math|SO(3)}} is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner.

The special orthogonal groups have additional [[Spin representation|spin representation]]s that are not tensor representations, and are ''typically'' not spherical harmonics. An exception are the [[Spin representation|spin representation]] of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the [[3-sphere|3-sphere]]. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication.

==Connection with hemispherical harmonics==
Spherical harmonics can be separated into two set of functions.<ref>{{Cite journal |vauthors=Zheng Y, Wei K, Liang B, Li Y, Chu X | date=2019-12-23 | title=Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering | url=https://www.osapublishing.org/oe/abstract.cfm?uri=oe-27-26-37180|journal=Optics Express | language=EN | volume=27 | issue=26 | pages=37180–37195 | doi=10.1364/OE.27.037180 | pmid=31878503 | bibcode=2019OExpr..2737180Z|issn=1094-4087|doi-access=free}}</ref> One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).

===Generalizations===
The [[Conformal geometry|angle-preserving symmetries]] of the two-sphere are described by the group of [[Möbius transformation]]s PSL(2,'''C'''). With respect to this group, the sphere is equivalent to the usual [[Riemann sphere]]. The group PSL(2,'''C''') is isomorphic to the (proper) [[Lorentz group]], and its action on the two-sphere agrees with the action of the Lorentz group on the [[Celestial sphere|celestial sphere]] in [[Physics:Minkowski space|Minkowski space]]. The analog of the spherical harmonics for the Lorentz group is given by the hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as {{math|1=SO(3) = PSU(2)}} is a [[Subgroup|subgroup]] of {{math|PSL(2,'''C''')}}.

More generally, hypergeometric series can be generalized to describe the symmetries of any [[Symmetric space|symmetric space]]; in particular, hypergeometric series can be developed for any [[Lie group]].<ref>N. Vilenkin, ''Special Functions and the Theory of Group Representations'', Am. Math. Soc. Transl., vol. 22, (1968).</ref><ref>J. D. Talman, ''Special Functions, A Group Theoretic Approach'', (based on lectures by E.P. Wigner), W. A. Benjamin, New York (1968).</ref><ref>W. Miller, ''Symmetry and Separation of Variables,'' Addison-Wesley, Reading (1977).</ref><ref>A. Wawrzyńczyk, ''Group Representations and Special Functions'', Polish Scientific Publishers. Warszawa (1984).</ref>

==See also==
{{cols}}
* [[Cubic harmonic]] (often used instead of spherical harmonics in computations)
* [[Cylindrical harmonics]]
* [[Spherical basis]]
* [[Spinor spherical harmonics]]
* [[Spin-weighted spherical harmonics]]
* [[Sturm–Liouville theory]]
* [[Table of spherical harmonics]]
* [[Vector spherical harmonics]]
* [[Physics:Atomic orbital|Atomic orbital]]
{{colend}}

==Notes==
<references />

==References==
=== Cited references ===
* {{Citation|first1=Richard|last1=Courant|first2=David|last2=Hilbert|author-link2=David Hilbert | title=Methods of Mathematical Physics, Volume I | publisher=Wiley-Interscience | year=1962}}.
* {{Citation|first=A.R.|last=Edmonds|title=Angular Momentum in Quantum Mechanics|year=1957|publisher=Princeton University Press | isbn=0-691-07912-9|url-access=registration|url=https://archive.org/details/angularmomentumi0000edmo}}
* {{Citation | last1=Eremenko | first1=Alexandre | last2=Jakobson | first2=Dmitry | last3=Nadirashvili | first3=Nikolai | title=On nodal sets and nodal domains on S² and R² |mr=2394544 | year=2007 | journal=[[Annales de l'Institut Fourier]] | issn=0373-0956 | volume=57 | issue=7 | pages=2345–2360 | doi=10.5802/aif.2335| url=http://www.numdam.org/item/AIF_2007__57_7_2345_0/ | doi-access=free }}
* {{citation|first=Brian C.|last=Hall|title=Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267 |publisher=Springer|year=2013| isbn=978-1461471158}}
* {{Citation|first=T.M.|last=MacRobert|title=Spherical harmonics: An elementary treatise on harmonic functions, with applications|year=1967|publisher=Pergamon Press}}.
* {{Citation|first1=Paul Herman Ernst|last1=Meijer|first2=Edmond|last2=Bauer|title=Group theory: The application to quantum mechanics|publisher=Dover|year=2004|isbn=978-0-486-43798-9}}.
* {{springer|title=Spherical harmonics|id=S/s086690|first=E.D.|last=Solomentsev|year=2001}}.
* {{Citation|last1=Stein|first1=Elias|first2=Guido|last2=Weiss|author-link2=Guido Weiss | title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press | year=1971 | isbn=978-0-691-08078-9 | location=Princeton, N.J.|url-access=registration|url=https://archive.org/details/introductiontofo0000stei}}.
* {{Citation|last=Unsöld|first=Albrecht|year=1927|title=Beiträge zur Quantenmechanik der Atome|journal=Annalen der Physik|volume=387|issue=3|pages=355–393|doi=10.1002/andp.19273870304|bibcode = 1927AnP...387..355U }}.
* {{Citation | last1=Whittaker | first1=E. T.| last2=Watson | first2=G. N.|author-link2=G. N. Watson | title=[[A Course of Modern Analysis]] | publisher={{wipe|Cambridge University Press}} | year=1927|page=392}}.

=== General references ===
* E.W. Hobson, ''The Theory of Spherical and Ellipsoidal Harmonics'', (1955) Chelsea Pub. Co., {{isbn|978-0-8284-0104-3}}.
* C. Müller, ''Spherical Harmonics'', (1966) Springer, Lecture Notes in Mathematics, Vol. 17, {{isbn|978-3-540-03600-5}}.
* E. U. Condon and G. H. Shortley, ''The Theory of Atomic Spectra'', (1970) Cambridge at the University Press, {{isbn|0-521-09209-4}}, ''See chapter 3''.
* J.D. Jackson, ''Classical Electrodynamics'', {{isbn|0-471-30932-X}}
* Albert Messiah, ''Quantum Mechanics'', volume II. (2000) Dover. {{isbn|0-486-40924-4}}.
* {{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | location=New York | isbn=978-0-521-88068-8 | chapter=Section 6.7. Spherical Harmonics | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=292}}
* D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii ''Quantum Theory of Angular Momentum'',(1988) World Scientific Publishing Co., Singapore, {{isbn|9971-5-0107-4}}
* {{MathWorld|title=Spherical harmonics|urlname=SphericalHarmonic}}
* {{Citation|first1=John|last1=Maddock|title=Spherical harmonics in Boost.Math|url=http://www.boost.org/doc/libs/1_66_0/libs/math/doc/html/math_toolkit/sf_poly/sph_harm.html}}

==External links==
* [http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonics] at [[MathWorld]]
* [https://www.quantum-physics.polytechnique.fr/sphericalHarmonics.php?lang=1 Spherical Harmonics 3D representation]

{{DEFAULTSORT:Spherical Harmonics}}
[[Category:Atomic physics]]
[[Category:Fourier analysis]]
[[Category:Harmonic analysis]]
[[Category:Partial differential equations]]
[[Category:Rotational symmetry]]
[[Category:Special hypergeometric functions]]

{{Sourceattribution|Spherical harmonics}}