Spherical Harmonic

The spherical harmonics $Y_l^m(\theta, \phi)$ are the angular portion of the solution to Laplace's Equation in Spherical Coordinates where azimuthal symmetry is not present. Some care must be taken in identifying the notational convention being used. In the below equations, $\theta$ is taken as the azimuthal (longitudinal) coordinate, and $\phi$ as the polar (latitudinal) coordinate (opposite the notation of Arfken 1985).

$$Y_l^m(\theta, \phi) \equiv \sqrt{{2l+1\over 4\pi} {(l-m)!\over (l+m)!}} P_l^m(\cos \phi)e^{im\theta},$$ (1)

where $m = -l$, $-1+1$, ..., 0, ..., $l$ and the normalization is chosen such that

$$\int^{2\pi}_0 \int^\pi_0 Y_l^m{Y_{l'}^{m'}}^*\sin \phi \,d\p... ...'}^{m'}}^*\,d(\cos \phi )\,d\theta = \delta_{mm'}\delta_{ll'},$$ (2)

where $\delta_{mn}$ is the Kronecker Delta. Sometimes, the Condon-Shortley Phase $(-1)^m$ is prepended to the definition of the spherical harmonics.

Integrals of the spherical harmonics are given by

$\int Y_{l_1}^{m_1}Y_{l_2}^{m_2}Y_{l_3}^{m_3}\,d\Omega$
$= \sqrt{(2l_1+1)(2l_2+1)(2l_3+1)\over 4\pi}\pmatrix{l_1 & l_2 & l_3\cr 0 & 0 & 0\cr} \pmatrix{l_1 & l_2 & l_3\cr m_1 & m_2 & m_3\cr},\quad$	(3)

where $\pmatrix{l_1 & l_2 & l_3\cr m_1 & m_2 & m_3\cr}$ is a Wigner 3j-Symbol (which is related to the Clebsch-Gordan Coefficients). The spherical harmonics obey

$$Y_l^{-l} = {1\over 2^ll!} \sqrt{(2l+1)!\over 4\pi} \,\sin^l\,\phi e^{-il\theta}$$ (4)

$$Y_l^0 = \sqrt{2l+1\over 4\pi} P_l(\cos \phi)$$ (5)

$$Y_l^{-m} = (-1)^m{Y_l^m}^*,$$ (6)

where $P_l(x)$ is a Legendre Polynomial.

The above illustrations show $\vert Y_l^m(\theta,\phi)\vert$ (top) and $\Re[Y_l^m(\theta,\phi)]$ and $\Im[Y_l^m(\theta,\phi)]$ (bottom). The first few spherical harmonics are

$$Y_0^0 = {1\over 2} {1\over\sqrt{\pi}}$$

$$Y_1^{-1} = {1\over 2} \sqrt{{3\over 2\pi}}\, \sin\phi \,e^{-i\theta}$$

$$Y_1^0 = {1\over 2} \sqrt{{3\over\pi}}\, \cos\phi$$

$$Y_1^1 = - {1\over 2} \sqrt{{3\over 2\pi}}\, \sin\phi \,e^{i\theta }$$

$$Y_2^{-2} = {1\over 4} \sqrt{{15\over 2\pi}}\, \sin^2\phi\, e^{-2i\theta}$$

$$Y_2^{-1} = {1\over 2} \sqrt{{15\over 2\pi}}\, \sin\phi \cos\phi \,e^{-i\theta}$$

$$Y_2^0 = {1\over 4} \sqrt{{5\over\pi}}\, (3\cos^2\phi-1)$$

$$Y_2^1 = - {1\over 2} \sqrt{{15\over 2\pi}}\, \sin\phi \cos\phi \,e^{i\theta}$$

$$Y_2^2 = {1\over 4} \sqrt{{15\over 2\pi}}\, \sin^2\phi\, e^{2i\theta}$$

$$Y_3^{-3} = {1\over 8} \sqrt{{35\over\pi}}\, \sin^3\phi\,e^{-3i\theta}$$

$$Y_3^{-2} = {1\over 4} \sqrt{{105\over 2\pi}}\, \sin^2\phi \cos\phi\, e^{-2i\theta}$$

$$Y_3^{-1} = {1\over 8} \sqrt{{21\over\pi}}\, \sin \phi(5\cos^2\phi-1)e^{-i\theta}$$

$$Y_3^0 = {1\over 4} \sqrt{{7\over\pi}}\, (5\cos^3\phi-3\cos\phi )$$

$$Y_3^1 = - {1\over 8} \sqrt{{21\over\pi}}\, \sin\phi(5\cos^2\phi-1)e^{i\theta}$$

$$Y_3^2 = {1\over 4} \sqrt{{105\over 2\pi}}\, \sin^2\phi \cos\phi \,e^{2i\theta}$$

$$Y_3^3 = - {1\over 8} \sqrt{{35\over\pi}}\, \sin^3\phi \,e^{3i\theta}.$$

Written in terms of Cartesian Coordinates,

$$e^{i\theta} = {x+iy\over\sqrt{x^2+y^2}}$$ (7)

$$\phi = \sin^{-1}\left({\sqrt{x^2+y^2\over x^2+y^2+z^2}\,}\right)$$ (8)

$$= \cos^{-1}\left({z\over\sqrt{x^2+y^2+z^2}}\right),$$ (9)

so

$$Y_0^0 = {1\over 2} {1\over\sqrt{\pi}}$$ (10)

$$Y_1^0 = {1\over 2}\sqrt{3\over\pi} {z\over\sqrt{x^2+y^2+z^2}}$$ (11)

$$Y_1^1 = -{1\over 2}\sqrt{3\over 2\pi} {x+iy\over\sqrt{x^2+y^2+z^2}}$$ (12)

$$Y_2^0 = {1\over 4}\sqrt{5\over\pi} \left({{3z^2\over x^2+y^2+z^2}-1}\right)$$ (13)

$$Y_2^1 = -{1\over 2}\sqrt{15\over 2\pi} {z(x+iy)\over x^2+y^2+z^2}$$ (14)

$$Y_2^2 = {1\over 4}\sqrt{15\over 2\pi} {(x+iy)^2\over x^2+y^2+z^2}.$$ (15)

These can be separated into their Real and Imaginary Parts

$${Y_l^m}^s(\theta, \phi) \equiv P_l^m(\cos\phi)\sin(m\theta)$$ (16)

$${Y_l^m}^c(\theta, \phi) \equiv P_l^m(\cos\phi)\cos(m\theta).$$ (17)

The Zonal Harmonics are defined to be those of the form

$$P_n^m(\cos\theta).$$ (18)

The Tesseral Harmonics are those of the form

$$\sin(m\phi)P_n^m(\cos\theta)$$ (19)

$$\cos(m\phi)P_n^m(\cos\theta)$$ (20)

for $n\not = m$. The Sectorial Harmonics are of the form

$$\sin(m\phi)P_m^m(\cos\theta)$$ (21)

$$\cos(m\phi)P_m^m(\cos\theta).$$ (22)

The spherical harmonics form a Complete Orthonormal Basis, so an arbitrary Real function $f(\theta, \phi)$ can be expanded in terms of Complex spherical harmonics

$$f(\theta, \phi)\equiv \sum_{l=0}^\infty \sum_{m=-1}^l A_l^m Y_l^m(\theta, \phi),$$ (23)

or Real spherical harmonics

$$f(\theta, \phi) \equiv \sum_{l=0}^\infty \sum_{m=0}^l [C_l^m... ...}^c(\theta, \phi)\sin(m\theta)+S_l^m {Y_l^m}^s(\theta, \phi)].$$ (24)

See also Correlation Coefficient, Spherical Harmonic Addition Theorem, Spherical Harmonic Closure Relations, Spherical Vector Harmonic

References

Spherical Harmonics

Arfken, G. ``Spherical Harmonics.'' §12.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 680-685, 1985.

Ferrers, N. M. An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them. London: Macmillan, 1877.

Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. New York: Cambridge University Press, 1996.

Hobson, E. W. The Theory of Spherical and Ellipsoidal Harmonics. New York: Chelsea, 1955.

MacRobert, T. M. and Sneddon, I. N. Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications, 3rd ed. rev. Oxford, England: Pergamon Press, 1967.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Spherical Harmonics.'' §6.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 246-248, 1992.

Sansone, G. ``Harmonic Polynomials and Spherical Harmonics,'' ``Integral Properties of Spherical Harmonics and the Addition Theorem for Legendre Polynomials,'' and ``Completeness of Spherical Harmonics with Respect to Square Integrable Functions.'' §3.18-3.20 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 253-272, 1991.

Sternberg, W. and Smith, T. L. The Theory of Potential and Spherical Harmonics, 2nd ed. Toronto: University of Toronto Press, 1946.