Spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations 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 and thus an 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 functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier 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 are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the 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 harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree in that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence from the abovementioned polynomial of degree ; the remaining factor can be regarded as a function of the spherical angular coordinates and only, or equivalently of the orientational unit vector 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 LaplaceBeltrami 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 LaplaceBeltrami operator (see the section Higher dimensions below).
A specific set of spherical harmonics, denoted or , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782.^{[1]} 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 electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D 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.
History[edit]
Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. In 1782, PierreSimon de Laplace had, in his Mécanique Céleste, determined that the gravitational potential at a point x associated with a set of point masses m_{i} located at points x_{i} was given by
Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time, AdrienMarie Legendre had investigated the expansion of the Newtonian potential in powers of r = x and r_{1} = x_{1}. He discovered that if r ≤ r_{1} then
where γ is the angle between the vectors x and x_{1}. The functions are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle γ between x_{1} and x. (See Applications of Legendre polynomials in physics for a more detailed analysis.)
In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. The solid harmonics were homogeneous polynomial solutions of Laplace's equation
The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complexvalued functions. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.
The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The (complexvalued) spherical harmonics are eigenfunctions of the square of the orbital angular momentum operator
Laplace's spherical harmonics[edit]
Laplace's equation imposes that the Laplacian of a scalar field f is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function .) In spherical coordinates this is:^{[2]}
Consider the problem of finding solutions of the form f(r, θ, φ) = R(r) Y(θ, φ). By separation of variables, two differential equations result by imposing Laplace's equation:
for some number m. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e^{± imφ}. The solution function Y(θ, φ) is regular at the poles of the sphere, where θ = 0, π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ (ℓ + 1) for some nonnegative integer with ℓ ≥ m; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial P^{m}
_{ℓ}(cos θ) . Finally, the equation for R has solutions of the form R(r) = A r^{ℓ} + B r^{−ℓ − 1}; requiring the solution to be regular throughout R^{3} forces B = 0.^{[3]}
Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of ℓ, there are 2ℓ + 1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:
which fulfill
Here is called a spherical harmonic function of degree ℓ and order m, is an associated Legendre polynomial, N is a normalization constant,^{[4]} and θ and φ represent colatitude and longitude, respectively. In particular, the colatitude θ, or polar angle, ranges from 0 at the North Pole, to π/2 at the Equator, to π at the South Pole, and the longitude φ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer ℓ, every solution Y(θ, φ), , of the eigenvalue problem
The general solution to Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r^{ℓ},
where the are constants and the factors r^{ℓ} Y_{ℓ}^{m} are known as (regular) solid harmonics . Such an expansion is valid in the ball
For , the solid harmonics with negative powers of (the irregular solid harmonics ) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about ), instead of the Taylor series (about ) used above, to match the terms and find series expansion coefficients .
Orbital angular momentum[edit]
In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum^{[5]}
These operators commute, and are densely defined selfadjoint operators on the weighted Hilbert space of functions f squareintegrable with respect to the normal distribution as the weight function on R^{3}:
If Y is a joint eigenfunction of L^{2} and L_{z}, then by definition
Denote this joint eigenspace by E_{λ,m}, and define the raising and lowering operators by
_{+} : E_{λ,m} → E_{λ,m+k} must be zero for k sufficiently large, because the inequality λ ≥ m^{2} must hold in each of the nontrivial joint eigenspaces. Let Y ∈ E_{λ,m} be a nonzero joint eigenfunction, and let k be the least integer such that
The foregoing has been all worked out in the spherical coordinate representation, but may be expressed more abstractly in the complete, orthonormal spherical ket basis.
Harmonic polynomial representation[edit]
The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions . Specifically, we say that a (complexvalued) polynomial function is homogeneous of degree if
For example, when , is just the 3dimensional space of all linear functions , since any such function is automatically harmonic. Meanwhile, when , we have a 5dimensional space:
For any , the space of spherical harmonics of degree is just the space of restrictions to the sphere of the elements of .^{[6]} As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a harmonic function).
For example, for any the formula
Conventions[edit]
Orthogonality and normalization[edit]
This section's factual accuracy is disputed. (December 2017) 
Several different normalizations are in common use for the Laplace spherical harmonic functions . Throughout the section, we use the standard convention that for (see associated Legendre polynomials)
In acoustics,^{[8]} the Laplace spherical harmonics are generally defined as (this is the convention used in this article)
where are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice).
In both definitions, the spherical harmonics are orthonormal
The disciplines of geodesy^{[11]} and spectral analysis use
which possess unit power
The magnetics^{[11]} community, in contrast, uses Schmidt seminormalized harmonics
which have the normalization
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
where the superscript * denotes complex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner Dmatrix.
Condon–Shortley phase[edit]
One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of , commonly referred to as the Condon–Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the 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 raising and lowering operators. The geodesy^{[12]} 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.^{[13]}
Real form[edit]
A real basis of spherical harmonics can be defined in terms of their complex analogues by setting
The real spherical harmonics are sometimes known as tesseral spherical harmonics.^{[14]} These functions have the same orthonormality properties as the complex ones above. The real spherical harmonics with m > 0 are said to be of cosine type, and those with m < 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.
See here for a list of real spherical harmonics up to and including , which can be seen to be consistent with the output of the equations above.
Use in quantum chemistry[edit]
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 nonrelativistic 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, the usual p functions () are complex and mix axis directions, but the real versions are essentially just x, y, and z.
Spherical harmonics in Cartesian form[edit]
The complex spherical harmonics give rise to the solid harmonics by extending from to all of as a homogeneous function of degree , i.e. setting
The Herglotz generating function[edit]
If the quantum mechanical convention is adopted for the , then
The essential property of is that it is null:
It suffices to take and as real parameters. In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, §VII.7, who credit unpublished notes by him for its discovery.
Essentially all the properties of the spherical harmonics can be derived from this generating function.^{[15]} An immediate benefit of this definition is that if the vector is replaced by the quantum mechanical spin vector operator , such that is the operator analogue of the solid harmonic ,^{[16]} one obtains a generating function for a standardized set of spherical tensor operators, :
The parallelism of the two definitions ensures that the 's transform under rotations (see below) in the same way as the 's, which in turn guarantees that they are spherical tensor operators, , with and , obeying all the properties of such operators, such as the ClebschGordan composition theorem, and the WignerEckart theorem. They are, moreover, a standardized set with a fixed scale or normalization.
Separated Cartesian form[edit]
The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of and another of and , as follows (Condon–Shortley phase):
The factor is essentially the associated Legendre polynomial , and the factors are essentially .
Examples[edit]
Using the expressions for , , and listed explicitly above we obtain:
Real forms[edit]
Using the equations above to form the real spherical harmonics, it is seen that for only the terms (cosines) are included, and for only the terms (sines) are included:
Special cases and values[edit]
 When , the spherical harmonics reduce to the ordinary Legendre polynomials:
 When , or more simply in Cartesian coordinates,
 At the north pole, where , and is undefined, all spherical harmonics except those with vanish:
Symmetry properties[edit]
The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.
Parity[edit]
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 . Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with being a unit vector,
In terms of the spherical angles, parity transforms a point with coordinates to . The statement of the parity of spherical harmonics is then
Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree ℓ changes the sign by a factor of (−1)^{ℓ}.
Rotations[edit]
Consider a rotation about the origin that sends the unit vector to . Under this operation, a spherical harmonic of degree and order transforms into a linear combination of spherical harmonics of the same degree. That is,
The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The 's of degree provide a basis set of functions for the irreducible representation of the group SO(3) of dimension . 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[edit]
The Laplace spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of squareintegrable functions . On the unit sphere , any squareintegrable function can thus be expanded as a linear combination of these:
This expansion holds in the sense of meansquare convergence — convergence in L^{2} of the sphere — which is to say that
The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:
If the coefficients decay in ℓ sufficiently rapidly — for instance, exponentially — then the series also converges uniformly to f.
A squareintegrable function can also be expanded in terms of the real harmonics above as a sum
The convergence of the series holds again in the same sense, namely the real spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of squareintegrable functions . The benefit of the expansion in terms of the real harmonic functions is that for real functions the expansion coefficients are guaranteed to be real, whereas their coefficients in their expansion in terms of the (considering them as functions ) do not have that property.
Spectrum analysis[edit]
This section needs additional citations for verification. (July 2020) 
Power spectrum in signal processing[edit]
The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. Using the orthonormality properties of the real unitpower 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 seminormalized harmonics, the relationship is slightly different for orthonormal harmonics):
is defined as the angular power spectrum (for Schmidt seminormalized harmonics). In a similar manner, one can define the crosspower of two functions as
is defined as the crosspower spectrum. If the functions f and g have a zero mean (i.e., the spectral coefficients f_{00} and g_{00} are zero), then S_{ff}(ℓ) and S_{fg}(ℓ) represent the contributions to the function's variance and covariance for degree ℓ, respectively. It is common that the (cross)power spectrum is well approximated by a power law of the form
When β = 0, the spectrum is "white" as each degree possesses equal power. When β < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when β > 0, the spectrum is termed "blue". The condition on the order of growth of S_{ff}(ℓ) is related to the order of differentiability of f in the next section.
Differentiability properties[edit]
One can also understand the differentiability properties of the original function f in terms of the asymptotics of S_{ff}(ℓ). In particular, if S_{ff}(ℓ) decays faster than any rational function of ℓ as ℓ → ∞, then f is infinitely differentiable. If, furthermore, S_{ff}(ℓ) decays exponentially, then f is actually real analytic on the sphere.
The general technique is to use the theory of Sobolev spaces. Statements relating the growth of the S_{ff}(ℓ) to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically, if
Algebraic properties[edit]
Addition theorem[edit]
A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. Given two vectors r and r′, with spherical coordinates and , respectively, the angle between them is given by the relation
The addition theorem states^{[17]}

(1) 
where P_{ℓ} is the Legendre polynomial of degree ℓ. This expression is valid for both real and complex harmonics.^{[18]} 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 zaxis, and then directly calculating the righthand side.^{[19]}
In particular, when x = y, this gives Unsöld's theorem^{[20]}
In the expansion (1), the lefthand side is a constant multiple of the degree ℓ zonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. Let Y_{j} be an arbitrary orthonormal basis of the space H_{ℓ} of degree ℓ spherical harmonics on the nsphere. Then , the degree ℓ zonal harmonic corresponding to the unit vector x, decomposes as^{[21]}

(2) 
Furthermore, the zonal harmonic is given as a constant multiple of the appropriate Gegenbauer polynomial:

(3) 
Combining (2) and (3) gives (1) in dimension n = 2 when x and y are represented in spherical coordinates. Finally, evaluating at x = y gives the functional identity
Contraction rule[edit]
Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics^{[22]}
Clebsch–Gordan coefficients[edit]
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 3jm symbol, the Racah coefficients, and the Slater integrals. Abstractly, the Clebsch–Gordan coefficients express the tensor product of two irreducible representations of the rotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.
Visualization of the spherical harmonics[edit]
The Laplace spherical harmonics can be visualized by considering their "nodal lines", that is, the set of points on the sphere where , or alternatively where . Nodal lines of are composed of ℓ circles: there are 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 in the and directions respectively. Considering as a function of , 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 as a function of , the trigonometric sin and cos functions possess 2m zeros, each of which gives rise to a nodal 'line of longitude'.
When the spherical harmonic order m is zero (upperleft in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Such spherical harmonics are a special case of zonal spherical functions. When ℓ = m (bottomright 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 ℓ are not necessarily those of the Laplace basis , and their nodal sets can be of a fairly general kind.^{[23]}
List of spherical harmonics[edit]
Analytic expressions for the first few orthonormalized Laplace spherical harmonics that use the Condon–Shortley phase convention:
Higher dimensions[edit]
The classical spherical harmonics are defined as complexvalued functions on the unit sphere inside threedimensional Euclidean space . Spherical harmonics can be generalized to higherdimensional Euclidean space as follows, leading to functions .^{[24]} Let P_{ℓ} denote the space of complexvalued homogeneous polynomials of degree ℓ in n real variables, here considered as functions . That is, a polynomial p is in P_{ℓ} provided that for any real , one has
Let A_{ℓ} denote the subspace of P_{ℓ} consisting of all harmonic polynomials:
The following properties hold:
 The sum of the spaces H_{ℓ} is dense in the set of continuous functions on with respect to the uniform topology, by the Stone–Weierstrass theorem. As a result, the sum of these spaces is also dense in the space L^{2}(S^{n−1}) of squareintegrable functions on the sphere. Thus every squareintegrable function on the sphere decomposes uniquely into a series of spherical harmonics, where the series converges in the L^{2} sense.
 For all f ∈ H_{ℓ}, one has where Δ_{Sn−1} is the Laplace–Beltrami operator on S^{n−1}. This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian in n dimensions decomposes as
 It follows from the Stokes theorem and the preceding property that the spaces H_{ℓ} are orthogonal with respect to the inner product from L^{2}(S^{n−1}). That is to say, for f ∈ H_{ℓ} and g ∈ H_{k} for k ≠ ℓ.
 Conversely, the spaces H_{ℓ} are precisely the eigenspaces of Δ_{Sn−1}. In particular, an application of the spectral theorem to the Riesz potential gives another proof that the spaces H_{ℓ} are pairwise orthogonal and complete in L^{2}(S^{n−1}).
 Every homogeneous polynomial p ∈ P_{ℓ} can be uniquely written in the form^{[25]} where p_{j} ∈ A_{j}. In particular,
An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the SturmLiouville problem for the spherical Laplacian
Connection with representation theory[edit]
The space H_{ℓ} of spherical harmonics of degree ℓ is a representation of the symmetry group of rotations around a point (SO(3)) and its doublecover SU(2). Indeed, rotations act on the twodimensional sphere, and thus also on H_{ℓ} by function composition
The elements of H_{ℓ} arise as the restrictions to the sphere of elements of A_{ℓ}: harmonic polynomials homogeneous of degree ℓ on threedimensional Euclidean space R^{3}. By polarization of ψ ∈ A_{ℓ}, there are coefficients symmetric on the indices, uniquely determined by the requirement
More generally, the analogous statements hold in higher dimensions: the space H_{ℓ} of spherical harmonics on the nsphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric ℓtensors. However, whereas every irreducible tensor representation of SO(2) and 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 representations that are not tensor representations, and are typically not spherical harmonics. An exception are the 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 3sphere. The spaces of spherical harmonics on the 3sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication.
Connection with hemispherical harmonics[edit]
Spherical harmonics can be separated into two set of functions.^{[28]} One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).
Generalizations[edit]
The anglepreserving symmetries of the twosphere are described by the group of Möbius transformations 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 twosphere agrees with the action of the Lorentz group on the celestial sphere in Minkowski space. The analog of the spherical harmonics for the Lorentz group is given by the hypergeometric series; furthermore, the spherical harmonics can be reexpressed in terms of the hypergeometric series, as SO(3) = PSU(2) is a subgroup of PSL(2,C).
More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group.^{[29]}^{[30]}^{[31]}^{[32]}
See also[edit]
 Cubic harmonic (often used instead of spherical harmonics in computations)
 Cylindrical harmonics
 Spherical basis
 Spinor spherical harmonics
 Spinweighted spherical harmonics
 Sturm–Liouville theory
 Table of spherical harmonics
 Vector spherical harmonics
 Atomic orbital
Notes[edit]
 ^ A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of MacRobert 1967. The term "Laplace spherical harmonics" is in common use; see Courant & Hilbert 1962 and Meijer & Bauer 2004.
 ^ The approach to spherical harmonics taken here is found in (Courant & Hilbert 1962, §V.8, §VII.5).
 ^ Physical applications often take the solution that vanishes at infinity, making A = 0. This does not affect the angular portion of the spherical harmonics.
 ^ Weisstein, Eric W. "Spherical Harmonic". mathworld.wolfram.com. Retrieved 20230510.
 ^ Edmonds 1957, §2.5
 ^ Hall 2013 Section 17.6
 ^ Hall 2013 Lemma 17.16
 ^ Williams, Earl G. (1999). Fourier acoustics : sound radiation and nearfield acoustical holography. San Diego, Calif.: Academic Press. ISBN 0080506909. OCLC 181010993.
 ^ Messiah, Albert (1999). Quantum mechanics : two volumes bound as one (Two vol. bound as one, unabridged reprint ed.). Mineola, NY: Dover. ISBN 9780486409245.
 ^ Claude CohenTannoudji; Bernard Diu; Franck Laloë (1996). Quantum mechanics. Translated by Susan Reid Hemley; et al. WileyInterscience: Wiley. ISBN 9780471569527.
 ^ ^{a} ^{b} Blakely, Richard (1995). Potential theory in gravity and magnetic applications. Cambridge England New York: Cambridge University Press. p. 113. ISBN 9780521415088.
 ^ Heiskanen and Moritz, Physical Geodesy, 1967, eq. 162
 ^ Weisstein, Eric W. "CondonShortley Phase". mathworld.wolfram.com. Retrieved 20221102.
 ^ Whittaker & Watson 1927, p. 392.
 ^ See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012).
 ^ Li, Feifei; Braun, Carol; Garg, Anupam (2013), "The WeylWignerMoyal Formalism for Spin", Europhysics Letters, 102 (6): 60006, arXiv:1210.4075, Bibcode:2013EL....10260006L, doi:10.1209/02955075/102/60006, S2CID 119610178
 ^ Edmonds, A. R. (1996). Angular Momentum In Quantum Mechanics. Princeton University Press. p. 63.
 ^ This is valid for any orthonormal basis of spherical harmonics of degree ℓ. For unit power harmonics it is necessary to remove the factor of 4π.
 ^ Whittaker & Watson 1927, p. 395
 ^ Unsöld 1927
 ^ Stein & Weiss 1971, §IV.2
 ^ Brink, D. M.; Satchler, G. R. Angular Momentum. Oxford University Press. p. 146.
 ^ Eremenko, Jakobson & Nadirashvili 2007
 ^ Solomentsev 2001; Stein & Weiss 1971, §Iv.2
 ^ Cf. Corollary 1.8 of Axler, Sheldon; Ramey, Wade (1995), Harmonic Polynomials and DirichletType Problems
 ^ Higuchi, Atsushi (1987). "Symmetric tensor spherical harmonics on the Nsphere and their application to the de Sitter group SO(N,1)". Journal of Mathematical Physics. 28 (7): 1553–1566. Bibcode:1987JMP....28.1553H. doi:10.1063/1.527513.
 ^ Hall 2013 Corollary 17.17
 ^ Zheng Y, Wei K, Liang B, Li Y, Chu X (20191223). "Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering". Optics Express. 27 (26): 37180–37195. Bibcode:2019OExpr..2737180Z. doi:10.1364/OE.27.037180. ISSN 10944087. PMID 31878503.
 ^ N. Vilenkin, Special Functions and the Theory of Group Representations, Am. Math. Soc. Transl., vol. 22, (1968).
 ^ J. D. Talman, Special Functions, A Group Theoretic Approach, (based on lectures by E.P. Wigner), W. A. Benjamin, New York (1968).
 ^ W. Miller, Symmetry and Separation of Variables, AddisonWesley, Reading (1977).
 ^ A. Wawrzyńczyk, Group Representations and Special Functions, Polish Scientific Publishers. Warszawa (1984).
References[edit]
Cited references[edit]
 Courant, Richard; Hilbert, David (1962), Methods of Mathematical Physics, Volume I, WileyInterscience.
 Edmonds, A.R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press, ISBN 0691079129
 Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai (2007), "On nodal sets and nodal domains on S² and R²", Annales de l'Institut Fourier, 57 (7): 2345–2360, doi:10.5802/aif.2335, ISSN 03730956, MR 2394544
 Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, ISBN 9781461471158
 MacRobert, T.M. (1967), Spherical harmonics: An elementary treatise on harmonic functions, with applications, Pergamon Press.
 Meijer, Paul Herman Ernst; Bauer, Edmond (2004), Group theory: The application to quantum mechanics, Dover, ISBN 9780486437989.
 Solomentsev, E.D. (2001) [1994], "Spherical harmonics", Encyclopedia of Mathematics, EMS Press.
 Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 9780691080789.
 Unsöld, Albrecht (1927), "Beiträge zur Quantenmechanik der Atome", Annalen der Physik, 387 (3): 355–393, Bibcode:1927AnP...387..355U, doi:10.1002/andp.19273870304.
 Whittaker, E. T.; Watson, G. N. (1927), A Course of Modern Analysis, Cambridge University Press, p. 392.
General references[edit]
 E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, (1955) Chelsea Pub. Co., ISBN 9780828401043.
 C. Müller, Spherical Harmonics, (1966) Springer, Lecture Notes in Mathematics, Vol. 17, ISBN 9783540036005.
 E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press, ISBN 0521092094, See chapter 3.
 J.D. Jackson, Classical Electrodynamics, ISBN 047130932X
 Albert Messiah, Quantum Mechanics, volume II. (2000) Dover. ISBN 0486409244.
 Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.7. Spherical Harmonics", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 9780521880688
 D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii Quantum Theory of Angular Momentum,(1988) World Scientific Publishing Co., Singapore, ISBN 9971501074
 Weisstein, Eric W. "Spherical harmonics". MathWorld.
 Maddock, John, Spherical harmonics in Boost.Math