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 that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle 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 SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
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. Functions that satisfy Laplace's equation are often said to be harmonic, hence the name spherical harmonics. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of 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.
A specific set of spherical harmonics, denoted or , are called 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, e.g., the representation of multipole electrostatic and electromagnetic fields, computation of atomic orbital electron configurations, representation of gravitational fields, geoids, fiber reconstruction for estimation of the path and location of neural axons based on the properties of water diffusion from diffusionweighted MRI imaging for streamline tractography, and the magnetic fields of planetary bodies and stars, and characterization of 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.
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 to 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 ? is the angle between the vectors x and x_{1}. The functions P_{i} are the Legendre polynomials, and they are a special case of spherical harmonics. Subsequently, in his 1782 memoire, Laplace investigated these coefficients using spherical coordinates to represent the angle ? 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
By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. (See the section below, "Harmonic polynomial representation".) The term "Laplace's coefficients" was employed by William Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had properly been introduced by Laplace and Legendre.
The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. 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 spherical harmonics are eigenfunctions of the square of the orbital angular momentum operator
and therefore they represent the different quantized configurations of atomic orbitals.
Laplace's equation imposes that the divergence of the gradient of a scalar field f is zero. In spherical coordinates this is:^{[2]}
Consider the problem of finding solutions of the form f(r, ?, ?) = R(r) Y(?, ?). By separation of variables, two differential equations result by imposing Laplace's equation:
The second equation can be simplified under the assumption that Y has the form Y(?, ?) = ?(?) ?(?). Applying separation of variables again to the second equation gives way to the pair of differential equations
for some number m. A priori, m is a complex constant, but because ? must be a periodic function whose period evenly divides 2?, m is necessarily an integer and ? is a linear combination of the complex exponentials e^{± im?}. The solution function Y(?, ?) is regular at the poles of the sphere, where ? = 0, ?. Imposing this regularity in the solution ? of the second equation at the boundary points of the domain is a SturmLiouville problem that forces the parameter ? to be of the form ? = 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 ? transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial P_{l}^{m}(cos ?) . 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(?, ?) = ?(?) ?(?). 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 Y_{l}^{m} is called a spherical harmonic function of degree l and order m, P_{l}^{m} is an associated Legendre polynomial, N is a normalization constant, and ? and ? represent colatitude and longitude, respectively. In particular, the colatitude ?, or polar angle, ranges from 0 at the North Pole, to ?/2 at the Equator, to ? at the South Pole, and the longitude ?, or azimuth, may assume all values with 0 ? < 2?. For a fixed integer l, every solution Y(?, ?) of the eigenvalue problem
is a linear combination of Y_{l}^{m}. In fact, for any such solution, r^{l} Y(?, ?) is the expression in spherical coordinates of a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are 2l + 1 linearly independent such polynomials.
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 f_{l}^{m} are constants and the factors r^{l} Y_{l}^{m} are known as solid harmonics. Such an expansion is valid in the ball
In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum^{[4]}
The ? is conventional in quantum mechanics; it is convenient to work in units in which ? = 1. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum
Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:
These operators commute, and are densely defined selfadjoint operators on the Hilbert space of functions f squareintegrable with respect to the normal distribution on R^{3}:
Furthermore, L^{2} is a positive operator.
If Y is a joint eigenfunction of L^{2} and L_{z}, then by definition
for some real numbers m and ?. Here m must in fact be an integer, for Y must be periodic in the coordinate ? with period a number that evenly divides 2?. Furthermore, since
and each of L_{x}, L_{y}, L_{z} are selfadjoint, it follows that ? >= m^{2}.
Denote this joint eigenspace by E_{?,m}, and define the raising and lowering operators by
Then L_{+} and L_{} commute with L^{2}, and the Lie algebra generated by L_{+}, L_{}, L_{z} is the special linear Lie algebra of order 2, , with commutation relations
Thus (it is a "raising operator") and (it is a "lowering operator"). In particular, must be zero for k sufficiently large, because the inequality ? >= m^{2} must hold in each of the nontrivial joint eigenspaces. Let Y ? E_{?,m} be a nonzero joint eigenfunction, and let k be the least integer such that
Then, since
it follows that
Thus ? = l(l+1) for the positive integer .
See also the section below on spherical harmonics in higher dimensions.
The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions on . Specifically, we say that a polynomial function on is homogeneous of degree if
for all real numbers and all . We say that is harmonic if
where is the Laplacian. Then for each , we define
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 .^{[5]} 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, the formula
defines a homogeneous polynomial of degree on , which happens to be independent of . This polynomial is easily seen to be harmonic. If we write in spherical coordinates and then restrict to , we obtain
which can be rewritten as
After using the formula for the associated Legendre polynomial , we may recognize this as the formula for the spherical harmonic ^{[6]} (See the section below on special cases of the spherical harmonics.)
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Several different normalizations are in common use for the Laplace spherical harmonic functions. Throughout the section, we use the standard convention that (see associated Legendre polynomials)
which is the natural normalization given by Rodrigues' formula.
In acoustics^{[7]}, the Laplace spherical harmonics are generally defined as (this is the convention used in this article)
while in quantum mechanics:^{[8]}^{[9]}
where are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice).
In both definitions, the spherical harmonics are orthonormal
where ?_{ij} is the Kronecker delta and d? = sin? d? d?. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.
The disciplines of geodesy^{[]} and spectral analysis use
which possess unit power
The magnetics^{[]} 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.
One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of (1)^{m}, commonly referred to as the CondonShortley 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 CondonShortley 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^{[10]} and magnetics communities never include the CondonShortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials.^{[]}
A real basis of spherical harmonics can be defined in terms of their complex analogues by setting
The CondonShortley phase convention is used here for consistency. The corresponding inverse equations are
The real spherical harmonics are sometimes known as tesseral spherical harmonics.^{[11]} These functions have the same orthonormality properties as the complex ones above. The 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.
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.
If the quantum mechanical convention is adopted for the , then,
Here, is the vector with components , and
is a vector with complex coefficients. It suffices to take as a real parameter. The essential property of is that it is null:
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.^{[12]} An immediate benefit of this definition is that if the cnumber vector is replaced by the quantum mechanical spin vector operator , 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.
The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of and another of and , as follows (CondonShortley phase):
and for m = 0:
Here
and
For this reduces to
The factor is essentially the associated Legendre polynomial , and the factors are essentially .
Using the expressions for , , and listed explicitly above we obtain:
It may be verified that this agrees with the function listed here and here.
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:
and for m = 0:
1. When , the spherical harmonics reduce to the ordinary Legendre polynomials:
2. When ,
or more simply in Cartesian coordinates,
3. At the north pole, where , and is undefined, all spherical harmonics except those with vanish:
The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.
The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. Inversion is represented by the operator . 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
(This can be seen as follows: The associated Legendre polynomials gives (1)^{l+m} and from the exponential function we have (1)^{m}, giving together for the spherical harmonics a parity of (1)^{l}.)
Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree l changes the sign by a factor of (1)^{l}.
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,
where is a matrix of order that depends on the rotation . However, this is not the standard way of expressing this property. In the standard way one writes,
where is the complex conjugate of an element of the Wigner Dmatrix.
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.
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 ?, 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 f can also be expanded in terms of the real harmonics Y_{lm} above as a sum
The convergence of the series holds again in the same sense, but the benefit of the real expansion is that for real functions f the expansion coefficients become real.
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):
where
is defined as the angular power spectrum (for Schmidt seminormalized harmonics). In a similar manner, one can define the crosspower of two functions as
where
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 ? = 0, the spectrum is "white" as each degree possesses equal power. When ? < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when ? > 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.
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
then f is in the Sobolev space H^{s}(S^{2}). In particular, the Sobolev embedding theorem implies that f is infinitely differentiable provided that
for all s.
A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. This is a generalization of the trigonometric identity
in which the role of the trigonometric functions appearing on the righthand side is played by the spherical harmonics and that of the lefthand side is played by the Legendre polynomials.
Consider two unit vectors x and y. The addition theorem states^{[13]}

where P_{l} is the Legendre polynomial of degree l. This expression is valid for both real and complex harmonics.^{[14]} 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.^{[15]}
In particular, when x = y, this gives Unsöld's theorem^{[16]}
which generalizes the identity cos^{2}? + sin^{2}? = 1 to two dimensions.
In the expansion (1), the lefthand side P_{l}(x·y) 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^{[17]}

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

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
where ?_{n1} is the volume of the (n1)sphere.
Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics ^{[18]}
where the values of and are determined by the selection rules for the 3jsymbols.
The ClebschGordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics itself. 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 ClebschGordan 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.
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 circles: some are latitudes and others are longitudes. One can determine the number of nodal lines of each type by counting the number of zeros of in the latitudinal and longitudinal directions independently. For the latitudinal direction,^{[clarification needed]} the real and imaginary components of the associated Legendre polynomials each possess lm zeros, whereas for the longitudinal direction, the trigonometric sin and cos functions possess 2m zeros.^{[clarification needed]}
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.^{[19]}
Analytic expressions for the first few orthonormalized Laplace spherical harmonics that use the CondonShortley phase convention:
The classical spherical harmonics are defined as functions on the unit sphere S^{2} inside threedimensional Euclidean space. Spherical harmonics can be generalized to higherdimensional Euclidean space R^{n} as follows.^{[20]} Let P_{l} denote the space of homogeneous polynomials of degree l in n variables. That is, a polynomial P is in P_{l} provided that
Let A_{l} denote the subspace of P_{l} consisting of all harmonic polynomials; these are the solid spherical harmonics. Let H_{l} denote the space of functions on the unit sphere
obtained by restriction from A_{l}.
The following properties hold:
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
where ? is the axial coordinate in a spherical coordinate system on S^{n1}. The end result of such a procedure is^{[21]}
where the indices satisfy l_{1} 2 n1 and the eigenvalue is l_{n1}(l_{n1} + n2). The functions in the product are defined in terms of the Legendre function
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
for ? a spherical harmonic and ? a rotation. The representation H_{l} is an irreducible representation of SO(3).^{[22]}
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 ? ? A_{l}, there are coefficients symmetric on the indices, uniquely determined by the requirement
The condition that ? be harmonic is equivalent to the assertion that the tensor must be trace free on every pair of indices. Thus as an irreducible representation of SO(3), H_{l} is isomorphic to the space of traceless symmetric tensors of degree l.
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.
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.^{[23]}^{[24]}^{[25]}^{[26]}