Spherical Harmonic

The spherical harmonics 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 this entry, is taken as the polar (colatitudinal) coordinate with , and as the azimuthal (longitudinal) coordinate with . This is the convention normally used in physics, as described by Arfken (1985) and the Wolfram Language (in mathematical literature, usually denotes the longitudinal coordinate and the colatitudinal coordinate). Spherical harmonics are implemented in the Wolfram Language as SphericalHarmonicY[l, m, theta, phi].

Spherical harmonics satisfy the spherical harmonic differential equation, which is given by the angular part of Laplace's equation in spherical coordinates. Writing in this equation gives

(1)

Multiplying by gives

(2)

Using separation of variables by equating the -dependent portion to a constant gives

(3)

which has solutions

(4)

Plugging in (3) into (2) gives the equation for the -dependent portion, whose solution is

(5)

where , , ..., 0, ..., , and is an associated Legendre polynomial. The spherical harmonics are then defined by combining and ,

(6)

where the normalization is chosen such that

(7)

(Arfken 1985, p. 681). Here, denotes the complex conjugate and is the Kronecker delta. Sometimes (e.g., Arfken 1985), the Condon-Shortley phase is prepended to the definition of the spherical harmonics.

The spherical harmonics are sometimes separated into their real and imaginary parts,

(8)

(9)

The spherical harmonics obey

			(10)
			(11)
			(12)

where is a Legendre polynomial.

Integrals of the spherical harmonics are given by

(13)

where is a Wigner 3j-symbol (which is related to the Clebsch-Gordan coefficients). Special cases include

			(14)
			(15)
			(16)
			(17)

(Arfken 1985, p. 700).

The above illustrations show (top), (bottom left), and (bottom right). The first few spherical harmonics are

			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)
			(28)
			(29)
			(30)
			(31)
			(32)
			(33)

Written in terms of Cartesian coordinates,

			(34)
			(35)
			(36)

so

			(37)
			(38)
			(39)
			(40)
			(41)
			(42)

The zonal harmonics are defined to be those of the form

(43)

The tesseral harmonics are those of the form

(44)

(45)

for . The sectorial harmonics are of the form

(46)

(47)