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.
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 originates from solving Laplace's equation in the spherical domains. Functions that solve 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 formula_1 in formula_2 that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence formula_3 from the above-mentioned polynomial of degree formula_1; the remaining factor can be regarded as a function of the spherical angular coordinates formula_5 and formula_6 only, or equivalently of the orientational unit vector formula_7 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 formula_8 or formula_9, are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. 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.
Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his "Mécanique Céleste", determined that the gravitational potential formula_10 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, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of "r" = |x| and "r"1 = |x1|. He discovered that if "r" ≤ "r"1 then
where γ is the angle between the vectors x and x1. The functions formula_13 are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. Subsequently, in his 1782 memoire, Laplace investigated these coefficients using spherical coordinates to represent the angle γ between x1 and x. (See Applications of Legendre polynomials in physics for a more detailed analysis.)
In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their "Treatise on Natural Philosophy", and also first introduced the name of "spherical harmonics" for these functions. The solid harmonics were homogeneous polynomial solutions formula_10 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. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.
The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The (complex-valued) spherical harmonics formula_16 are eigenfunctions of the square of the orbital angular momentum operator
and therefore they represent the different quantized configurations of atomic orbitals.
Laplace's spherical harmonics.
Laplace's equation imposes that the Laplacian of a scalar field is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function formula_18.) In spherical coordinates this is:
Consider the problem of finding solutions of the form . By separation of variables, two differential equations result by imposing Laplace's equation:
The second equation can be simplified under the assumption that has the form . Applying separation of variables again to the second equation gives way to the pair of differential equations
for some number . A priori, is a complex constant, but because must be a periodic function whose period evenly divides , is necessarily an integer and is a linear combination of the complex exponentials . The solution function is regular at the poles of the sphere, where . Imposing this regularity in the solution of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter to be of the form for some non-negative integer with ; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial . Finally, the equation for has solutions of the form ; requiring the solution to be regular throughout forces .
Here the solution was assumed to have the special form . For a given value of , there are independent solutions of this form, one for each integer with . These angular solutions formula_23 are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:
which fulfill
Here formula_26 is called a spherical harmonic function of degree and order , formula_27 is an associated Legendre polynomial, is a normalization constant, and and represent colatitude and longitude, respectively. In particular, the colatitude , or polar angle, ranges from at the North Pole, to at the Equator, to at the South Pole, and the longitude , or azimuth, may assume all values with . For a fixed integer , every solution , formula_28, of the eigenvalue problem
is a linear combination of formula_30. In fact, for any such solution, is the expression in spherical coordinates of a homogeneous polynomial formula_31 that is harmonic (see below), and so counting dimensions shows that there are linearly independent such polynomials.
The general solution formula_18 to Laplace's equation formula_33 in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor ,
where the formula_35 are constants and the factors are known as ("regular") solid harmonics formula_31. Such an expansion is valid in the ball
For formula_38, the solid harmonics with negative powers of formula_39 (the "irregular" solid harmonics formula_40) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about formula_41), instead of the Taylor series (about formula_42) used above, to match the terms and find series expansion coefficients formula_43.
Orbital angular momentum.
In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum
The is conventional in quantum mechanics; it is convenient to work in units in which . 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 self-adjoint operators on the weighted Hilbert space of functions "f" square-integrable with respect to the normal distribution as the weight function on R3:
Furthermore, L2 is a positive operator.
If "Y" is a joint eigenfunction of L2 and "L""z", then by definition
for some real numbers "m" and λ. Here "m" must in fact be an integer, for "Y" must be periodic in the coordinate φ with period a number that evenly divides 2π. Furthermore, since
and each of "L""x", "L""y", "L""z" are self-adjoint, it follows that λ ≥ "m"2.
Denote this joint eigenspace by "E"λ,"m", and define the raising and lowering operators by
Then "L"+ and "L"− commute with L2, and the Lie algebra generated by "L"+, "L"−, "L""z" is the special linear Lie algebra of order 2, formula_51, 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 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
Then, since
it follows that
Thus λ = ℓ(ℓ+1) for the positive integer .
The foregoing has been all worked out in the spherical coordinate representation, formula_56 but may be expressed more abstractly in the complete, orthonormal spherical ket basis.
Harmonic polynomial representation.
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 formula_57. Specifically, we say that a (complex-valued) polynomial function formula_58 is "homogeneous" of degree formula_1 if 
for all real numbers formula_61 and all formula_62. We say that formula_63 is "harmonic" if
where formula_65 is the Laplacian. Then for each formula_1, we define
For example, when formula_68, formula_69 is just the 3-dimensional space of all linear functions formula_31, since any such function is automatically harmonic. Meanwhile, when formula_71, we have a 5-dimensional space:
For any formula_1, the space formula_74 of spherical harmonics of degree formula_1 is just the space of restrictions to the sphere formula_76 of the elements of formula_77. 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 formula_78 the formula
defines a homogeneous polynomial of degree formula_1 with domain and codomain formula_57, which happens to be independent of formula_82. This polynomial is easily seen to be harmonic. If we write formula_63 in spherical coordinates formula_84 and then restrict to formula_85, we obtain
which can be rewritten as
After using the formula for the associated Legendre polynomial formula_88, we may recognize this as the formula for the spherical harmonic formula_89 (See the section below on special cases of the spherical harmonics.)
Conventions.
Orthogonality and normalization.
Several different normalizations are in common use for the Laplace spherical harmonic functions formula_16. Throughout the section, we use the standard convention that for formula_91 (see associated Legendre polynomials)
which is the natural normalization given by Rodrigues' formula.
In acoustics, the Laplace spherical harmonics are generally defined as (this is the convention used in this article)
while in quantum mechanics: 
where formula_95 are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice).
In both definitions, the spherical harmonics are orthonormal
where δ"ij" is the Kronecker delta and "d"Ω = sin θ "d"φ "d"θ. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.
The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics formula_98 in terms of the real spherical harmonics formula_99 are
The real spherical harmonics formula_99 are sometimes known as "tesseral spherical harmonics". These functions have the same orthonormality properties as the complex ones formula_98 above.
The real spherical harmonics formula_103 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
formula_104
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 formula_105, 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, the usual "p" functions (formula_106) are complex and mix axis directions, but the real versions are essentially just "x", "y" and "z".
Spherical harmonics in Cartesian form.
The Herglotz generating function.
If the quantum mechanical convention is adopted for the formula_107, then
Here, formula_7 is the vector with components formula_110, formula_111, and
is a vector with complex coefficients. It suffices to take formula_113 and formula_114 as real parameters.
The essential property of formula_115 is that it is null:
In naming this generating function after Herglotz, we follow , who credit unpublished notes by him for its discovery.
Essentially all the properties of the spherical harmonics can be derived from this generating function. An immediate benefit of this definition is that if the vector
formula_7 is replaced by the quantum mechanical spin vector operator formula_118, such that formula_119 is the operator analogue of the solid harmonic formula_120, one obtains a generating function for a standardized set of spherical tensor operators,
formula_119:
The parallelism of the two definitions ensures that the formula_123's transform under rotations (see below) in the same way as the formula_124's, which in turn guarantees that they are spherical tensor operators, formula_125, with formula_126 and formula_127, 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 formula_128 and another of formula_129 and formula_130, as follows (Condon–Shortley phase):
and for "m" = 0:
Here
and
For formula_136 this reduces to
The factor formula_138 is essentially the associated Legendre polynomial formula_139, and the factors formula_140 are essentially formula_141.
Examples.
Using the expressions for formula_142, formula_143, and formula_144 listed explicitly above we obtain: 
It may be verified that this agrees with the function listed here and here.
Real forms.
Using the equations above to form the real spherical harmonics, it is seen that for formula_91 only the formula_148 terms (cosines) are included, and for formula_149 only the formula_150 terms (sines) are included:
and for "m" = 0:
Special cases and values.
1. When formula_136, the spherical harmonics formula_107 reduce to the ordinary Legendre polynomials:
2. When formula_156,
or more simply in Cartesian coordinates,
3. At the north pole, where formula_159, and formula_6 is undefined,
all spherical harmonics except those with formula_161 vanish:
Symmetry properties.
The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.
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 formula_163. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with
formula_164 being a unit vector,
In terms of the spherical angles, parity transforms a point with coordinates formula_166 to formula_167. 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.
Consider a rotation formula_169 about the origin that sends the unit vector formula_164 to formula_171. Under this operation, a spherical harmonic of degree formula_1 and order formula_173 transforms into a linear combination of spherical harmonics of the same degree. That is,
where formula_175 is a matrix of order formula_176 that depends on the
rotation formula_169. However, this is not the standard way of expressing this property. In the standard way one writes,
where formula_179 is the complex conjugate of an element of the Wigner D-matrix. In particular when formula_180 is a formula_181 rotation of the azimuth we get the identity,
The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The formula_183's of degree formula_1 provide a basis set of functions for the irreducible representation of the group SO(3) of dimension formula_176. 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 formula_26 form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions formula_187. On the unit sphere formula_76, any square-integrable function formula_189 can thus be expanded as a linear combination of these:
This expansion holds in the sense of mean-square convergence — convergence in L2 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 square-integrable function formula_193 can also be expanded in terms of the real harmonics formula_99 above as a sum
The convergence of the series holds again in the same sense, namely the real spherical harmonics formula_99 form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions formula_197. The benefit of the expansion in terms of the real harmonic functions formula_103 is that for real functions formula_193 the expansion coefficients formula_200 are guaranteed to be real, whereas their coefficients formula_201 in their expansion in terms of the formula_124 (considering them as functions formula_203) do not have that property.
Harmonical tensors.
Formula.
As a rule, harmonic functions are useful in theoretical physics to consider fields in the far field when the distance from charges is much farther than the size of their location. In this case, the radius R is constant and coordinates (θ,φ) are convenient to use. Theoretical physics considers many problems when a solution of Laplace's equation is needed as a function of Сartesian coordinates. At the same time, it is important to get an invariant form of solutions relative to the rotation of space or, generally speaking, relative to group transformations.
The simplest tensor solutions – dipole, quadrupole and octupole potentials – are fundamental concepts of general physics:
It is easy to verify that they are the harmonic functions. The total set of tensors is defined by the Taylor series of a point charge field potential for formula_207:
where tensor formula_209 is denoted by symbol formula_210 and contraction of the tensors is in the brackets [...].
Therefore, the tensor formula_211 is defined by the formula_212-th tensor derivative:
James Clerk Maxwell used similar considerations without tensors naturally. E. W. Hobson analysed Maxwell's method as well.
One can see from the equation the following properties that repeat mainly those of solid and spherical functions.
The formula for a harmonic invariant tensor was found in paper. A detailed description is given in the monography.
The formula contains products of tensors formula_223 and Kronecker symbols formula_224:
The number of Kronecker symbols is increased by two in the product of each following item when the range of tensors formula_226 is reduced by two accordingly. The operation formula_227 symmetrizes a tensor by means of summing all "independent" permutations of indices. Particularly, each formula_228 does not need to be transformed into formula_229 and tensors formula_230 do not become formula_231.
These tensors are convenient to substitute into Laplace's equation:
The last relation is Euler's formula for homogeneous polynomials. The Laplace operator formula_233 does not affect the index symmetry of tensors. The two relations allow substitution of a tensor into Laplace's equation to check directly that the tensor is a harmonic function:
Simplified moments.
The last property is important for theoretical physics for the following reason. Potential of charges outside of their location is integral to be equal to the sum of multipole potentials:
where formula_236 is the charge density.
The convolution is applied to tensors in the formula naturally. Integrals in the sum are called in physics as multipole moments. Three of them are used actively while others applied less often as their structure (or that of spherical functions) is more complicated. Nevertheless, last property gives the way to simplify calculations in theoretical physics by using integrals with tensor formula_237 instead of harmonical tensor formula_219. Therefore, simplified moments give the same result and there is no need to restrict calculations for dipole, quadrupole and octupole potentials only. It is the advantage of the tensor point of view and not the only that.
Efimov's ladder operator.
Spherical functions have a few recurrent formulas. In quantum mechanics recurrent formulas plays a role when they connect formula_239functions of quantum states by means of a ladder operator.The property is occurred due to symmetry group of considered system. The vector ladder operator for the invariant harmonical states found in paper and detailed in. 
Operator formula_242 applying to the harmonical tensor potential in formula_243-space goes into Efimov's ladder operator acting on transformed tensor in formula_244-space:
where formula_246 is operator of module of angular momentum:
Operator formula_248 multiplies harmonic tensor by its degree i.e. by formula_212 if to recall according spherical function for quantum numbers formula_212, formula_173.
To check action of the ladder operator formula_252, one can apply it to dipole and quadrupole tensors:
Applying successively formula_252 to formula_256 we get general form of invariant harmonic tensors:
The operator formula_252 analogous to the oscillator ladder operator. To trace relation with a quantum operator it is useful to multiply it by formula_259 to go to reversed space:
As a result, operator goes into the operator of momentum in formula_261-space :
It is useful to apply the following properties of formula_252 .
The property is utterly convenient for calculations.
The property gives zero trace of the harmonical tensor formula_266 over each two indices.
The ladder operator is analogous for that in problem of the quantum oscillator. It generates Glauber states those are created in the quantum theory of electromagnetic radiation fields.
It was shown later as theoretical result that the coherent states are intrinsic for any quantum system with a group symmetry to include the rotational group.
Invariant form of spherical harmonics.
Spherical harmonics accord with the system of coordinates. Let be formula_267 the unit vectors along axises X, Y, Z. Denote following unit vectors as formula_268 and formula_269:
Using the vectors, the solid harmonics are equal to:
where formula_273 is the constant:
Angular momentum formula_276 is defined by the rotational group. The mechanical momentum formula_277 is related to the translation group. The ladder operator is the mapping of momentum upon inversion 1/r of 3-d space. It is raising operator. Lowering operator here is the gradient naturally together with partial contraction over pair indices formula_278 to leave others:
Spectrum analysis.
Power spectrum in signal processing.
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 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):
where
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
where
is defined as the cross-power 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.
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".
Algebraic properties.
Addition theorem.
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 formula_287 and formula_288, respectively, the angle formula_289 between them is given by the relation
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
where "P"ℓ is the Legendre polynomial of degree ℓ. This expression is valid for both real and complex harmonics. 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.
In particular, when x = y, this gives Unsöld's theorem
which generalizes the identity cos2θ + sin2θ = 1 to two dimensions.
In the expansion (), the left-hand 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 Hℓ of degree ℓ spherical harmonics on the "n"-sphere. Then formula_292, the degree ℓ zonal harmonic corresponding to the unit vector "x", decomposes as
Furthermore, the zonal harmonic formula_293 is given as a constant multiple of the appropriate Gegenbauer polynomial:
Combining () and () gives () in dimension "n" = 2 when x and y are represented in spherical coordinates. Finally, evaluating at x = y gives the functional identity
where ω"n"−1 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
where the values of formula_296 and formula_289 are determined by the selection rules for the 3j-symbols.
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 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.
The Laplace spherical harmonics formula_183 can be visualized by considering their "nodal lines", that is, the set of points on the sphere where formula_299, or alternatively where formula_300. Nodal lines of formula_183 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 formula_183 in the formula_5 and formula_6 directions respectively. Considering formula_183 as a function of formula_5, 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 formula_183 as a function of formula_6, the trigonometric sin and cos functions possess 2|"m"| zeros, each of which gives rise to a nodal 'line of longitude'.
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. Such spherical harmonics are a special case of zonal spherical functions. When ℓ = |"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 ℓ are not necessarily those of the Laplace basis formula_183, and their nodal sets can be of a fairly general kind.
List of spherical harmonics.
Analytic expressions for the first few orthonormalized Laplace spherical harmonics formula_98 that use the Condon–Shortley phase convention:
Higher dimensions.
The classical spherical harmonics are defined as complex-valued functions on the unit sphere formula_76 inside three-dimensional Euclidean space formula_321. Spherical harmonics can be generalized to higher-dimensional Euclidean space formula_322 as follows, leading to functions formula_323. Let Pℓ denote the space of complex-valued homogeneous polynomials of degree ℓ in "n" real variables, here considered as functions formula_324. That is, a polynomial "p" is in Pℓ provided that for any real formula_61, one has
Let Aℓ denote the subspace of Pℓ consisting of all harmonic polynomials:
These are the (regular) solid spherical harmonics. Let Hℓ denote the space of functions on the unit sphere
obtained by restriction from Aℓ
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 Sturm-Liouville problem for the spherical Laplacian
where φ is the axial coordinate in a spherical coordinate system on "S""n"−1. The end result of such a procedure is
where the indices satisfy |ℓ1| ≤ ℓ2 ≤ ... ≤ ℓ"n"−1 and the eigenvalue is −ℓ"n"−1(ℓ"n"−1 + "n"−2). The functions in the product are defined in terms of the Legendre function
Connection with representation theory.
The space Hℓ of spherical harmonics of degree ℓ is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). Indeed, rotations act on the two-dimensional sphere, and thus also on Hℓ by function composition
for ψ a spherical harmonic and ρ a rotation. The representation Hℓ is an irreducible representation of SO(3).
The elements of Hℓ arise as the restrictions to the sphere of elements of Aℓ: harmonic polynomials homogeneous of degree ℓ on three-dimensional Euclidean space R3. By polarization of ψ ∈ Aℓ, there are coefficients formula_340 symmetric on the indices, uniquely determined by the requirement
The condition that ψ be harmonic is equivalent to the assertion that the tensor formula_340 must be trace free on every pair of indices. Thus as an irreducible representation of SO(3), Hℓ is isomorphic to the space of traceless symmetric tensors of degree ℓ.
More generally, the analogous statements hold in higher dimensions: the space Hℓ of spherical harmonics on the "n"-sphere is the irreducible representation of SO("n"+1) corresponding to the traceless symmetric ℓ-tensors. 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 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. One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).
Generalizations.
The angle-preserving symmetries of the two-sphere 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 two-sphere 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 re-expressed 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.