Editing Spherical harmonics (section)

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

In [[mathematics]] and [[Outline of physical science|physical science]], '''spherical harmonics''' are [[special function]]s defined on the surface of a [[sphere]]. They are often employed in solving [[partial differential equation]]s in many scientific fields. The [[table of spherical harmonics]] contains a list of common spherical harmonics.

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

Spherical harmonics originate from solving [[Laplace's equation]] in the spherical domains. Functions that are solutions to Laplace's equation are called [[Harmonic function|harmonics]]. Despite their name, spherical harmonics take their simplest form in [[Cartesian coordinates]], where they can be defined as [[homogeneous polynomial]]s of [[Degree of a polynomial|degree]] <math>\ell</math> in <math>(x, y, z)</math> that obey Laplace's equation. The connection with [[Spherical coordinate system|spherical coordinates]] arises immediately if one uses the homogeneity to extract a factor of radial dependence <math>r^\ell</math> from the above-mentioned polynomial of degree <math>\ell</math>; the remaining factor can be regarded as a function of the spherical angular coordinates <math>\theta</math> and <math>\varphi</math> only, or equivalently of the [[Orientation (geometry)|orientational]] [[unit vector]] <math>\mathbf r</math> specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics are ''not'' functions on the sphere which are harmonic with respect to the [[Laplace-Beltrami operator]]  for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the [[Maximum principle]]. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see [[#Higher dimensions|Higher dimensions]]).

A specific set of spherical harmonics, denoted <math>Y_\ell^m(\theta,\varphi)</math> or <math>Y_\ell^m({\mathbf r})</math>, are known as Laplace's spherical harmonics, as they were first introduced by [[Pierre Simon de Laplace]] in 1782.<ref>A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of {{harvnb|MacRobert|1967}}. The term "Laplace spherical harmonics" is in common use; see {{harvnb|Courant|Hilbert|1962}} and {{harvnb|Meijer|Bauer|2004}}.</ref> These functions form an [[orthogonal]] system, and are thus basic to the expansion of a general function on the sphere as alluded to above.

Spherical harmonics are important in many theoretical and practical applications, including the representation of [[Multipole expansion|multipole]] electrostatic and [[electromagnetic field]]s, [[electron configuration]]s, [[gravitational field]]s, [[geoid]]s, the [[magnetic field]]s 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|precomputed radiance transfer]], etc.) and modelling of 3D shapes.