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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physical science Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences". Definition Phys ...
, spherical harmonics are special functions defined on the surface of a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of
orthogonal functions In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the ...
and thus an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
, 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 In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
, 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 polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s of degree \ell in (x, y, z) that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence r^\ell from the above-mentioned polynomial of degree \ell; the remaining factor can be regarded as a function of the spherical angular coordinates \theta and \varphi only, or equivalently of the orientational unit vector \mathbf r specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics are ''not'' functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the
Maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section ''Higher dimensions'' below). A specific set of spherical harmonics, denoted Y_\ell^m(\theta,\varphi) or Y_\ell^m(), 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 A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
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 3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...
, spherical harmonics play a role in a wide variety of topics including indirect lighting ( ambient occlusion,
global illumination Global illumination (GI), or indirect illumination, is a group of algorithms used in 3D computer graphics that are meant to add more realistic lighting to 3D scenes. Such algorithms take into account not only the light that comes directly from ...
,
precomputed radiance transfer Precomputed Radiance Transfer (PRT) is a computer graphics technique used to render a scene in real time with complex light interactions being precomputed to save time. Radiosity methods can be used to determine the diffuse lighting of the scene, ...
, etc.) and modelling of 3D shapes.


History

Spherical harmonics were first investigated in connection with the
Newtonian potential In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object ...
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 \R^3 \to \R at a point associated with a set of point masses located at points was given by V(\mathbf) = \sum_i \frac. 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 and . He discovered that if then \frac = P_0(\cos\gamma)\frac + P_1(\cos\gamma)\frac + P_2(\cos\gamma)\frac+\cdots where is the angle between the vectors and . The functions P_i:
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\to \R 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 between and . (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 ''Treatise on Natural Philosophy'' was an 1867 text book by William Thomson (later Lord Kelvin) and Peter Guthrie Tait, published by Oxford University Press. The ''Treatise'' was often referred to as T and ''T^1'', as explained by Alexander Ma ...
'', and also first introduced the name of “spherical harmonics” for these functions. The
solid harmonics In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), wh ...
were homogeneous polynomial solutions \R^3 \to \R of Laplace's equation \frac + \frac + \frac = 0. 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 William Whewell ( ; 24 May 17946 March 1866) was an English polymath, scientist, Anglican priest, philosopher, theologian, and historian of science. He was Master of Trinity College, Cambridge. In his time as a student there, he achieved ...
to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the
zonal spherical harmonics In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notio ...
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 The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
. 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 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
, 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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. The (complex-valued) spherical harmonics S^2 \to \Complex are eigenfunctions of the square of the orbital angular momentum operator -i\hbar\mathbf\times\nabla, and therefore they represent the different quantized configurations of
atomic orbitals In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any sp ...
.


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 f:\R^3 \to \Complex.) In spherical coordinates this is: \nabla^2 f = \frac \frac\left(r^2 \frac\right) + \frac \frac\left(\sin\theta \frac\right) + \frac \frac = 0. Consider the problem of finding solutions of the form . By
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
, two differential equations result by imposing Laplace's equation: \frac\frac\left(r^2\frac\right) = \lambda,\qquad \frac\frac\frac\left(\sin\theta \frac\right) + \frac\frac\frac = -\lambda. 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 \frac \frac = -m^2 \lambda\sin^2\theta + \frac \frac \left(\sin\theta \frac\right) = m^2 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
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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 Y_^m : S^2 \to \Complex are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Y_\ell^m (\theta, \varphi ) = N e^ P_\ell^m (\cos ) which fulfill r^2\nabla^2 Y_\ell^m (\theta, \varphi ) = -\ell (\ell + 1 ) Y_\ell^m (\theta, \varphi ). Here Y_^m:S^2 \to \Complex is called a spherical harmonic function of degree and order , P_^m: 1,1to \R 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 Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
, or
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematical ...
, may assume all values with . For a fixed integer , every solution , Y: S^2 \to \Complex, of the eigenvalue problem r^2\nabla^2 Y = -\ell (\ell + 1 ) Y is a linear combination of Y_\ell^m : S^2 \to \Complex. In fact, for any such solution, is the expression in spherical coordinates of a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
\R^3 \to \Complex that is harmonic (see
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), and so counting dimensions shows that there are linearly independent such polynomials. The general solution f:\R^3 \to \Complex to Laplace's equation \Delta f = 0 in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor , f(r, \theta, \varphi) = \sum_^\infty \sum_^\ell f_\ell^m r^\ell Y_\ell^m (\theta, \varphi ), where the f_^m \in \Complex are constants and the factors are known as (''regular'')
solid harmonics In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), wh ...
\R^3 \to \Complex. Such an expansion is valid in the ball r < R = \frac. For r > R, the solid harmonics with negative powers of r (the ''irregular''
solid harmonics In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), wh ...
\R^3 \setminus \ \to \Complex) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about r=\infty), instead of the Taylor series (about r = 0) used above, to match the terms and find series expansion coefficients f^m_\ell \in \Complex.


Orbital angular momentum

In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum \mathbf = -i\hbar (\mathbf\times \mathbf) = L_x\mathbf + L_y\mathbf+L_z\mathbf. 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 \begin \mathbf^2 &= -r^2\nabla^2 + \left(r\frac+1\right)r\frac\\ &= -\frac \frac\sin\theta \frac - \frac \frac. \end 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: \begin L_z &= -i\left(x\frac - y\frac\right)\\ &=-i\frac. \end These operators commute, and are densely defined self-adjoint operators on the
weighted A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
Hilbert space of functions ''f'' square-integrable with respect to the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
as the weight function on R3: \frac\int_ , f(x), ^2 e^\,dx < \infty. Furthermore, L2 is a positive operator. If is a joint eigenfunction of and , then by definition \begin \mathbf^2Y &= \lambda Y\\ L_zY &= mY \end for some real numbers ''m'' and ''λ''. Here ''m'' must in fact be an integer, for ''Y'' must be periodic in the coordinate ''φ'' with period a number that evenly divides 2''π''. Furthermore, since \mathbf^2 = L_x^2 + L_y^2 + L_z^2 and each of ''L''''x'', ''L''''y'', ''L''''z'' are self-adjoint, it follows that . Denote this joint eigenspace by , and define the raising and lowering operators by \begin L_+ &= L_x + iL_y\\ L_- &= L_x - iL_y \end Then and commute with , and the Lie algebra generated by , , is the special linear Lie algebra of order 2, \mathfrak_2(\Complex), with commutation relations _z,L_+= L_+,\quad _z,L_-= -L_-, \quad _+,L_-= 2L_z. Thus (it is a "raising operator") and (it is a "lowering operator"). In particular, must be zero for ''k'' sufficiently large, because the inequality must hold in each of the nontrivial joint eigenspaces. Let be a nonzero joint eigenfunction, and let be the least integer such that L_+^kY = 0. Then, since L_-L_+ = \mathbf^2 - L_z^2 - L_z it follows that 0 = L_-L_+^k Y = (\lambda - (m+k)^2-(m+k))Y. Thus for the positive integer . The foregoing has been all worked out in the spherical coordinate representation, \langle \theta, \varphi, l m\rangle = Y_l^m (\theta, \varphi) but may be expressed more abstractly in the complete, orthonormal spherical ket basis.


Harmonic polynomial representation

The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions \R^3 \to \Complex. Specifically, we say that a (complex-valued) polynomial function p: \R^3 \to \Complex is ''homogeneous'' of degree \ell if p(\lambda\mathbf x)=\lambda^\ell p(\mathbf x) for all real numbers \lambda \in \R and all x \in \R^3. We say that p is ''harmonic'' if \Delta p=0, where \Delta is the Laplacian. Then for each \ell, we define \mathbf_\ell = \left\. For example, when \ell=1, \mathbf_1 is just the 3-dimensional space of all linear functions \R^3 \to \Complex, since any such function is automatically harmonic. Meanwhile, when \ell = 2, we have a 5-dimensional space: \mathbf_2 = \operatorname_(x_1 x_2,\, x_1 x_3,\, x_2 x_3,\, x_1^2-x_2^2,\, x_1^2-x_3^2). For any \ell, the space \mathbf_ of spherical harmonics of degree \ell is just the space of restrictions to the sphere S^2 of the elements of \mathbf_\ell. 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 c \in \Complex the formula p(x_1, x_2, x_3) = c(x_1 + ix_2)^\ell defines a homogeneous polynomial of degree \ell with domain and codomain \R^3 \to \Complex, which happens to be independent of x_3. This polynomial is easily seen to be harmonic. If we write p in spherical coordinates (r,\theta,\varphi) and then restrict to r = 1, we obtain p(\theta,\varphi) = c \sin(\theta)^\ell (\cos(\varphi) + i \sin(\varphi))^\ell, which can be rewritten as p(\theta,\varphi) = c\left(\sqrt\right)^\ell e^. After using the formula for the associated Legendre polynomial P^\ell_\ell, we may recognize this as the formula for the spherical harmonic Y^\ell_\ell(\theta, \varphi). (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 S^2 \to \Complex. Throughout the section, we use the standard convention that for m>0 (see associated Legendre polynomials) P_\ell ^ = (-1)^m \frac P_\ell ^ 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) Y_\ell^m( \theta , \varphi ) = \sqrt \, P_\ell^m ( \cos ) \, e^ while in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
: Y_\ell^m( \theta , \varphi ) = (-1)^m \sqrt \, P_^m ( \cos ) \, e^ where P_^ are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice). In both definitions, the spherical harmonics are orthonormal \int_^\pi\int_^Y_\ell^m \, Y_^^* \, d\Omega=\delta_\, \delta_, where is the Kronecker delta and . This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e., \int = 1. The disciplines of geodesy and spectral analysis use Y_\ell^m( \theta , \varphi ) = \sqrt \, P_\ell^m ( \cos )\, e^ which possess unit power \frac \int_^\pi\int_^Y_\ell^m \, Y_^^* d\Omega=\delta_\, \delta_. The magnetics community, in contrast, uses Schmidt semi-normalized harmonics Y_\ell^m( \theta , \varphi ) = \sqrt \, P_\ell^m ( \cos ) \, e^ which have the normalization \int_^\pi\int_^Y_\ell^m \, Y_^^*d\Omega = \frac \delta_\, \delta_. 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 Y_\ell^^* (\theta, \varphi) = (-1)^m Y_\ell^ (\theta, \varphi), where the superscript denotes complex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix.


Condon–Shortley phase

One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of (-1)^m, 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 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.


Real form

A real basis of spherical harmonics Y_:S^2 \to \R can be defined in terms of their complex analogues Y_^m: S^2 \to \Complex by setting \begin Y_ &= \begin \dfrac \left(Y_\ell^ - (-1)^m\, Y_\ell^\right) & \text\ m < 0\\ Y_\ell^0 & \text\ m=0\\ \dfrac \left(Y_\ell^ + (-1)^m\, Y_\ell^\right) & \text\ m > 0. \end\\ &= \begin \dfrac \left(Y_\ell^ - (-1)^\, Y_\ell^\right) & \text\ m < 0\\ Y_\ell^0 & \text\ m=0\\ \dfrac \left(Y_\ell^ + (-1)^\, Y_\ell^\right) & \text\ m>0. \end\\ &= \begin \sqrt \, (-1)^m \, \Im [] & \text\ m<0\\ Y_\ell^0 & \text\ m=0\\ \sqrt \, (-1)^m \, \Re [] & \text\ m>0. \end \end The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics Y_^m : S^2 \to \Complex in terms of the real spherical harmonics Y_:S^2 \to \R are Y_^ = \begin \dfrac \left(Y_ - i Y_\right) & \text\ m<0 \\ ptY_ &\text\ m=0 \\ pt\dfrac \left(Y_ + i Y_\right) & \text\ m>0. \end The real spherical harmonics Y_:S^2 \to \R are sometimes known as ''tesseral spherical harmonics''. These functions have the same orthonormality properties as the complex ones Y_^m : S^2 \to \Complex above. The real spherical harmonics Y_ with are said to be of cosine type, and those with of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as Y_ = \begin \left(-1\right)^m\sqrt \sqrt \; P_\ell^(\cos \theta) \ \sin( , m, \varphi ) &\text m<0 \\ pt \sqrt \ P_\ell^m(\cos \theta) & \text m=0 \\ pt \left(-1\right)^m\sqrt \sqrt \; P_\ell^m(\cos \theta) \ \cos( m\varphi ) & \text m>0 \,. \end The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. See
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for a list of real spherical harmonics up to and including \ell = 4, 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 This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree \ell = 10. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in ''x'', ''y'' ...
, the usual functions (\ell = 1) are complex and mix axis directions, but the real versions are essentially just , , and .


Spherical harmonics in Cartesian form

The complex spherical harmonics Y_\ell^m give rise to the
solid harmonics In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), wh ...
by extending from S^2 to all of \R^3 as a homogeneous function of degree \ell, i.e. setting R_\ell^m(v) := \, v\, ^\ell Y_\ell^m\left(\frac\right) It turns out that R_\ell^m is basis of the space of harmonic and
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s of degree \ell. More specifically, it is the (unique up to normalization) Gelfand-Tsetlin-basis of this representation of the rotational group SO(3) and an explicit formula for R_\ell^m in cartesian coordinates can be derived from that fact.


The Herglotz generating function

If the quantum mechanical convention is adopted for the Y_^m: S^2 \to \Complex, then e^ = \sum_^ \sum_^ \sqrt \frac Y_^m (\mathbf/r). Here, \mathbf r is the vector with components (x, y, z) \in \R^3, r = , \mathbf, , and = - \frac\left( + i \right) + \frac\left( - i \right). \mathbf a is a vector with complex coordinates: \mathbf a = frac(\frac-\lambda),-\frac(\frac +\lambda),1 . The essential property of \mathbf a is that it is null: \mathbf a \cdot \mathbf a = 0. It suffices to take v and \lambda as real parameters. 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 \mathbf r is replaced by the quantum mechanical spin vector operator \mathbf J, such that \mathcal_^m() is the operator analogue of the solid harmonic r^Y_^m (\mathbf/r), one obtains a generating function for a standardized set of
spherical tensor operator In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of th ...
s, \mathcal_^m(): e^ = \sum_^ \sum_^ \sqrt \frac _^m(). The parallelism of the two definitions ensures that the \mathcal_^m's transform under rotations (see below) in the same way as the Y_^m's, which in turn guarantees that they are spherical tensor operators, T^_q, with k = and q = m, 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 z and another of x and y, as follows (Condon–Shortley phase): r^\ell\, \begin Y_\ell^ \\ Y_\ell^ \end = \left frac\right \bar^m_\ell(z) \begin \left(-1\right)^m (A_m + i B_m) \\ (A_m - i B_m) \end , \qquad m > 0. and for : r^\ell\,Y_\ell^ \equiv \sqrt \bar^0_\ell . Here A_m(x,y) = \sum_^m \binom x^p y^ \cos \left((m-p) \frac\right), B_m(x,y) = \sum_^m \binom x^p y^ \sin \left((m-p) \frac\right), and \bar^m_\ell(z) = \left frac\right \sum_^ (-1)^k 2^ \binom\binom \frac \; r^\; z^. For m = 0 this reduces to \bar^0_\ell(z) = \sum_^ (-1)^k 2^ \binom\binom \; r^\; z^. The factor \bar_\ell^m(z) is essentially the associated Legendre polynomial P_\ell^m(\cos\theta), and the factors (A_m \pm i B_m) are essentially e^.


Examples

Using the expressions for \bar_\ell^m(z), A_m(x,y), and B_m(x,y) listed explicitly above we obtain: Y^1_3 = - \frac \left tfrac\cdot \tfrac \right \left(5z^2-r^2\right) \left(x+iy\right) = - \left tfrac\cdot \tfrac\right \left(5\cos^2\theta-1\right) \left(\sin\theta e^\right) Y^_4 = \frac \left tfrac\cdot\tfrac\right \left(7z^2-r^2\right) \left(x-iy\right)^2 = \left tfrac\cdot\tfrac\right \left(7 \cos^2\theta -1\right) \left(\sin^2\theta e^\right) It may be verified that this agrees with the function listed
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and
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.


Real forms

Using the equations above to form the real spherical harmonics, it is seen that for m>0 only the A_m terms (cosines) are included, and for m<0 only the B_m terms (sines) are included: r^\ell\, \begin Y_ \\ Y_ \end = \sqrt \bar^m_\ell(z) \begin A_m \\ B_m \end , \qquad m > 0. and for ''m'' = 0: r^\ell\,Y_ \equiv \sqrt \bar^0_\ell .


Special cases and values

# When m = 0, the spherical harmonics Y_^m: S^2 \to \Complex reduce to the ordinary Legendre polynomials: Y_^0(\theta, \varphi) = \sqrt P_(\cos\theta). # When m = \pm\ell, Y_^(\theta,\varphi) = \frac \sqrt \sin^\theta\, e^, or more simply in Cartesian coordinates, r^ Y_^() = \frac \sqrt (x \pm i y)^. # At the north pole, where \theta = 0, and \varphi is undefined, all spherical harmonics except those with m = 0 vanish: Y_^m(0,\varphi) = Y_^m() = \sqrt \delta_.


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 P\Psi(\mathbf r) = \Psi(-\mathbf r). Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with \mathbf r being a unit vector, Y_\ell^m(-\mathbf r) = (-1)^\ell Y_\ell^m(\mathbf r). In terms of the spherical angles, parity transforms a point with coordinates \ to \. The statement of the parity of spherical harmonics is then Y_\ell^m(\theta,\varphi) \to Y_\ell^m(\pi-\theta,\pi+\varphi) = (-1)^\ell Y_\ell^m(\theta,\varphi) (This can be seen as follows: The associated Legendre polynomials gives and from the exponential function we have , giving together for the spherical harmonics a parity of .) 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 changes the sign by a factor of .


Rotations

Consider a rotation \mathcal R about the origin that sends the unit vector \mathbf r to \mathbf r'. Under this operation, a spherical harmonic of degree \ell and order m transforms into a linear combination of spherical harmonics of the same degree. That is, Y_\ell^m(') = \sum_^\ell A_ Y_\ell^(), where A_ is a matrix of order (2\ell + 1) that depends on the rotation \mathcal R. However, this is not the standard way of expressing this property. In the standard way one writes, Y_\ell^m(') = \sum_^\ell ^_()* Y_\ell^(), where D^_()^* is the complex conjugate of an element of the Wigner D-matrix. In particular when \mathbf r' is a \phi_0 rotation of the azimuth we get the identity, Y_\ell^m(') = Y_\ell^() e^. The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The Y_\ell^m's of degree \ell provide a basis set of functions for the irreducible representation of the group SO(3) of dimension (2\ell + 1). Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.


Spherical harmonics expansion

The Laplace spherical harmonics Y_^m:S^2 \to \Complex form a complete set of orthonormal functions and thus form an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of the Hilbert space of square-integrable functions L^2_(S^2). On the unit sphere S^2, any square-integrable function f:S^2 \to \Complex can thus be expanded as a linear combination of these: f(\theta,\varphi)=\sum_^\infty \sum_^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi). This expansion holds in the sense of mean-square convergence — convergence in L2 of the sphere — which is to say that \lim_ \int_0^\int_0^\pi \left, f(\theta,\varphi)-\sum_^N \sum_^\ell f_\ell^m Y_\ell^m(\theta,\varphi)\^2\sin\theta\, d\theta \,d\varphi = 0. 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: f_\ell^m=\int_ f(\theta,\varphi)\, Y_\ell^(\theta,\varphi)\,d\Omega = \int_0^d\varphi\int_0^\pi \,d\theta\,\sin\theta f(\theta,\varphi)Y_\ell^ (\theta,\varphi). If the coefficients decay in ''ℓ'' sufficiently rapidly — for instance, exponentially — then the series also
converges uniformly In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
to ''f''. A square-integrable function f:S^2 \to \R can also be expanded in terms of the real harmonics Y_:S^2 \to \R above as a sum f(\theta, \varphi) = \sum_^\infty \sum_^\ell f_ \, Y_(\theta, \varphi). The convergence of the series holds again in the same sense, namely the real spherical harmonics Y_:S^2 \to \R form a complete set of orthonormal functions and thus form an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of the Hilbert space of square-integrable functions L^2_(S^2). The benefit of the expansion in terms of the real harmonic functions Y_ is that for real functions f:S^2 \to \R the expansion coefficients f_ are guaranteed to be real, whereas their coefficients f_^m in their expansion in terms of the Y_^m (considering them as functions f: S^2 \to \Complex \supset \R) do not have that property.


Spectrum analysis


Power spectrum in signal processing

The total power of a function ''f'' is defined in the
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
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): \frac \int_\Omega , f(\Omega), ^2\, d\Omega = \sum_^\infty S_(\ell), where S_(\ell) = \frac\sum_^\ell , f_, ^2 is defined as the angular power spectrum (for Schmidt semi-normalized harmonics). In a similar manner, one can define the cross-power of two functions as \frac \int_\Omega f(\Omega) \, g^\ast(\Omega) \, d\Omega = \sum_^\infty S_(\ell), where S_(\ell) = \frac\sum_^\ell f_ g^\ast_ is defined as the cross-power spectrum. If the functions and have a zero mean (i.e., the spectral coefficients and are zero), then and represent the contributions to the function's variance and covariance for degree , respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form S_(\ell) = C \, \ell^. When , the spectrum is "white" as each degree possesses equal power. When , the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when , the spectrum is termed "blue". The condition on the order of growth of is related to the order of differentiability of in the next section.


Differentiability properties

One can also understand the differentiability properties of the original function in terms of the asymptotics of . In particular, if decays faster than any rational function of as , then is infinitely differentiable. If, furthermore, decays exponentially, then is actually
real analytic In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...
on the sphere. The general technique is to use the theory of
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s. Statements relating the growth of the to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically, if \sum_^\infty (1+\ell^2)^s S_(\ell) < \infty, then is in the Sobolev space . In particular, the
Sobolev embedding theorem In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the ...
implies that is infinitely differentiable provided that S_(\ell) = O(\ell^)\quad\rm\ell\to\infty for all .


Algebraic properties


Addition theorem

A mathematical result of considerable interest and use is called the ''addition theorem'' for spherical harmonics. Given two vectors and , with spherical coordinates (r,\theta,\varphi) and (r ', \theta ', \varphi '), respectively, the angle \gamma between them is given by the relation \cos\gamma = \cos\theta'\cos\theta + \sin\theta\sin\theta' \cos(\varphi-\varphi') 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 is the Legendre polynomial of degree . 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 , this gives Unsöld's theorem \sum_^\ell Y_^*(\mathbf) \, Y_(\mathbf) = \frac which generalizes the identity to two dimensions. In the expansion (), the left-hand side is a constant multiple of the degree
zonal spherical harmonic In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notio ...
. From this perspective, one has the following generalization to higher dimensions. Let be an arbitrary orthonormal basis of the space of degree spherical harmonics on the -sphere. Then Z^_, the degree zonal harmonic corresponding to the unit vector , decomposes as Furthermore, the zonal harmonic Z^_() is given as a constant multiple of the appropriate
Gegenbauer polynomial In mathematics, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are orthogonal polynomials on the interval minus;1,1with respect to the weight function (1 − ''x''2)''α''–1/2. They generalize Legendre polynom ...
: Combining () and () gives () in dimension when and are represented in spherical coordinates. Finally, evaluating at gives the functional identity \frac = \sum_^, Y_j(), ^2 where 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 Y_\left(\theta,\varphi\right)Y_\left(\theta,\varphi\right) = \sqrt\sum_^\sum_^\left(-1\right)^\sqrt\begin a & b & c\\ \alpha & \beta & -\gamma \end \begin a & b & c\\ 0 & 0 & 0 \end Y_\left(\theta,\varphi\right). Many of the terms in this sum are trivially zero. The values of c and \gamma that result in non-zero terms in this sum 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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
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 Y_\ell^m can be visualized by considering their " nodal lines", that is, the set of points on the sphere where \Re _\ell^m= 0, or alternatively where \Im _\ell^m= 0. Nodal lines of Y_\ell^m are composed of ''ℓ'' circles: there are 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 Y_\ell^m in the \theta and \varphi directions respectively. Considering Y_\ell^m as a function of \theta, 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 Y_\ell^m as a function of \varphi, 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 function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant v ...
s. When (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 are not necessarily those of the Laplace basis Y_\ell^m, 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 Y_^m : S^2 \to \Complex that use the Condon–Shortley phase convention: Y_^(\theta,\varphi) = \frac\sqrt \begin Y_^(\theta,\varphi) &= \frac\sqrt \, \sin\theta \, e^ \\ Y_^(\theta,\varphi) &= \frac\sqrt\, \cos\theta \\ Y_^(\theta,\varphi) &= \frac\sqrt\, \sin\theta\, e^ \end \begin Y_^(\theta,\varphi) &= \frac\sqrt \, \sin^\theta \, e^ \\ Y_^(\theta,\varphi) &= \frac\sqrt\, \sin\theta\, \cos\theta\, e^ \\ Y_^(\theta,\varphi) &= \frac \sqrt\, (3\cos^\theta-1) \\ Y_^(\theta,\varphi) &= \frac\sqrt\, \sin\theta\,\cos\theta\, e^ \\ Y_^(\theta,\varphi) &= \frac\sqrt\, \sin^\theta \, e^ \end


Higher dimensions

The classical spherical harmonics are defined as complex-valued functions on the unit sphere S^2 inside three-dimensional Euclidean space \R^3. Spherical harmonics can be generalized to higher-dimensional Euclidean space \R^n as follows, leading to functions S^ \to \Complex. Let P''ℓ'' denote the
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
of complex-valued
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s of degree in real variables, here considered as functions \R^n \to \Complex. That is, a polynomial is in provided that for any real \lambda \in \R, one has p(\lambda \mathbf) = \lambda^\ell p(\mathbf). Let A''ℓ'' denote the subspace of P''ℓ'' consisting of all
harmonic polynomial In mathematics, in abstract algebra, a multivariate polynomial over a field such that the Laplacian of is zero is termed a harmonic polynomial. The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In ...
s: \mathbf_ := \ \,. These are the (regular) solid spherical harmonics. Let H''ℓ'' denote the space of functions on the unit sphere S^ := \ obtained by restriction from \mathbf_ := \left\ . The following properties hold: * The sum of the spaces is dense in the set C(S^) of continuous functions on S^ with respect to the uniform topology, by the
Stone–Weierstrass theorem In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the ...
. As a result, the sum of these spaces is also dense in the space of square-integrable functions on the sphere. Thus every square-integrable function on the sphere decomposes uniquely into a series of spherical harmonics, where the series converges in the sense. * For all , one has \Delta_f = -\ell(\ell+n-2)f. where is the Laplace–Beltrami operator on . This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian in dimensions decomposes as \nabla^2 = r^\fracr^\frac + r^\Delta_ = \frac + \frac\frac + r^\Delta_ * It follows from the
Stokes theorem In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms o ...
and the preceding property that the spaces are orthogonal with respect to the inner product from . That is to say, \int_ f\bar \, \mathrm\Omega = 0 for and for . * Conversely, the spaces are precisely the eigenspaces of . In particular, an application of the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
to the Riesz potential \Delta_^ gives another proof that the spaces are pairwise orthogonal and complete in . * Every homogeneous polynomial can be uniquely written in the form p(x) = p_\ell(x) + , x, ^2p_ + \cdots + \begin , x, ^\ell p_0 & \ell \rm\\ , x, ^ p_1(x) & \ell\rm \end where . In particular, \dim \mathbf_\ell = \binom-\binom. An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
, by solving the Sturm-Liouville problem for the spherical Laplacian \Delta_ = \sin^\varphi\frac\sin^\varphi\frac + \sin^\varphi \Delta_ where ''φ'' is the axial coordinate in a spherical coordinate system on ''S''''n''−1. The end result of such a procedure is Y_ (\theta_1, \dots \theta_) = \frac e^ \prod_^ _j \bar^_ (\theta_j) where the indices satisfy and the eigenvalue is . The functions in the product are defined in terms of the Legendre function _j \bar^\ell_ (\theta) = \sqrt \sin^ (\theta) P^_ (\cos \theta) \,.


Connection with representation theory

The space of spherical harmonics of degree 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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
, and thus also on by function composition \psi \mapsto \psi\circ\rho^ for a spherical harmonic and a rotation. The representation is an irreducible representation of SO(3). The elements of arise as the restrictions to the sphere of elements of : harmonic polynomials homogeneous of degree on three-dimensional Euclidean space . By polarization of , there are coefficients \psi_ symmetric on the indices, uniquely determined by the requirement \psi(x_1,\dots,x_n) = \sum_\psi_x_\cdots x_. The condition that be harmonic is equivalent to the assertion that the tensor \psi_ must be trace free on every pair of indices. Thus as an irreducible representation of , is isomorphic to the space of traceless symmetric tensors of degree . More generally, the analogous statements hold in higher dimensions: the space of spherical harmonics on the -sphere is the irreducible representation of corresponding to the traceless symmetric -tensors. However, whereas every irreducible tensor representation of and 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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
. 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 is a subgroup of . 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.A. Wawrzyńczyk, ''Group Representations and Special Functions'', Polish Scientific Publishers. Warszawa (1984).


See also

* Cubic harmonic (often used instead of spherical harmonics in computations) *
Cylindrical harmonics In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, \nabla^2 V = 0, expressed in cylindrical coordinates, ''ρ'' (radial coordinate), ''φ'' (polar angle), a ...
*
Spherical basis In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular m ...
* Spinor spherical harmonics *
Spin-weighted spherical harmonics In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, t ...
* Sturm–Liouville theory *
Table of spherical harmonics This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree \ell = 10. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in ''x'', ''y'' ...
* Vector spherical harmonics *
Atomic orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...


Notes


References


Cited references

* . * * * * . * . * . * . * . * .


General references

* E.W. Hobson, ''The Theory of Spherical and Ellipsoidal Harmonics'', (1955) Chelsea Pub. Co., . * C. Müller, ''Spherical Harmonics'', (1966) Springer, Lecture Notes in Mathematics, Vol. 17, . * E. U. Condon and G. H. Shortley, ''The Theory of Atomic Spectra'', (1970) Cambridge at the University Press, , ''See chapter 3''. * J.D. Jackson, ''Classical Electrodynamics'', * Albert Messiah, ''Quantum Mechanics'', volume II. (2000) Dover. . * * D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii ''Quantum Theory of Angular Momentum'',(1988) World Scientific Publishing Co., Singapore, * * {{DEFAULTSORT:Spherical Harmonics Atomic physics Fourier analysis Harmonic analysis Partial differential equations Rotational symmetry Special hypergeometric functions


External links


Spherical Harmonics
at MathWorld
Spherical Harmonics 3D representation