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. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). P ℓ ( x ⋅ y ) = 4 π 2 ℓ + 1 ∑ m = − ℓ ℓ Y ℓ m ( y ) Y ℓ m ∗ ( x ) ∀ ℓ ∈ N 0 ∀ x , y ∈ R 2 : ‖ x ‖ 2 = ‖ y ‖ 2 = 1 , {displaystyle P_{ell }(mathbf {x} cdot mathbf {y} )={frac {4pi }{2ell +1}}sum _{m=-ell }^{ell }Y_{ell m}(mathbf {y} ),Y_{ell m}^{*}(mathbf {x} )quad forall ,ell in mathbb {N} _{0};forall ,mathbf {x} ,mathbf {y} in mathbb {R} ^{2}colon ;|mathbf {x} |_{2}=|mathbf {y} |_{2}=1,,}     (1) Z x ( ℓ ) ( y ) = ∑ j = 1 dim ⁡ ( H ℓ ) Y j ( x ) ¯ Y j ( y ) {displaystyle Z_{mathbf {x} }^{(ell )}({mathbf {y} })=sum _{j=1}^{dim(mathbf {H} _{ell })}{overline {Y_{j}({mathbf {x} })}},Y_{j}({mathbf {y} })}     (2) Z x ( ℓ ) ( y ) = C ℓ ( ( n − 1 ) / 2 ) ( x ⋅ y ) {displaystyle Z_{mathbf {x} }^{(ell )}({mathbf {y} })=C_{ell }^{((n-1)/2)}({mathbf {x} }cdot {mathbf {y} })}     (3) 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. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree ℓ {displaystyle ell } in ( x , y , z ) {displaystyle (x,y,z)} that obey Laplace's equation. Functions that satisfy Laplace's equation are often said to be harmonic, hence the name spherical harmonics. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of r ℓ {displaystyle r^{ell }} from the above-mentioned polynomial of degree ℓ {displaystyle ell } ; the remaining factor can be regarded as a function of the spherical angular coordinates θ {displaystyle heta } and φ {displaystyle varphi } only, or equivalently of the orientational unit vector r {displaystyle {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. A specific set of spherical harmonics, denoted Y ℓ m ( θ , φ ) {displaystyle Y_{ell }^{m}( heta ,varphi )} or Y ℓ m ( r ) {displaystyle Y_{ell }^{m}({mathbf {r} })} , are called Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functionsform an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. Spherical harmonics are important in many theoretical and practical applications, e.g., the representation of multipole electrostatic and electromagnetic fields, computation of atomic orbital electron configurations, representation of gravitational fields, geoids, fiber reconstruction for estimation of the path and location of neural axons based on the properties of water diffusion from diffusion-weighted MRI imaging for streamline tractography, and the magnetic fields of planetary bodies and stars, and characterization of the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes. Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his Mécanique Céleste, determined that the gravitational potential at a point x associated with a set of point masses mi located at points xi 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 r1 = |x1|. He discovered that if r ≤ r1 then where γ is the angle between the vectors x and x1. The functions Pi are the Legendre polynomials, and they are a special case of spherical harmonics. Subsequently, in his 1782 memoire, Laplace investigated these coefficients using spherical coordinates to represent the angle 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 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.