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The spherical harmonics can be generalized to vector spherical harmonics by looking for a scalar function and a constant vector such that(1)(2)(3)(4)so(5)Now interchange the order of differentiation and use the fact that multiplicative constants can be moving inside and outside the derivatives to obtain(6)(7)(8)and(9)(10)Putting these together gives(11)so satisfies the vector Helmholtz differential equation if satisfies the scalar Helmholtz differential equation(12)Construct another vector function(13)which also satisfies the vector Helmholtzdifferential equation since(14)(15)(16)(17)(18)which gives(19)We have the additional identity(20)(21)(22)(23)(24)In this formalism, is called the generating function and is called the pilot vector. The choice of generating function is determined by the symmetry of the scalar equation, i.e., it is chosen to solve the desired scalar differential equation. If is taken as(25)where..

A zonal harmonic is a spherical harmonic of the form , i.e., one which reduces to a Legendre polynomial (Whittaker and Watson 1990, p. 302). These harmonics are termed "zonal" since the curves on a unit sphere (with center at the origin) on which vanishes are parallels of latitude which divide the surface into zones (Whittaker and Watson 1990, p. 392).Resolving into factors linear in , multiplied by when is odd, then replacing by allows the zonal harmonic to be expressed as a product of factors linear in , , and , with the product multiplied by when is odd (Whittaker and Watson 1990, p. 1990).

A tesseral harmonic is a spherical harmonic of the form . These harmonics are so named because the curves on which they vanish are parallels of latitude and meridians, which divide the surface of a sphere into quadrangles whose angles are right angles (Whittaker and Watson 1990, p. 392).Resolving into factors linear in , multiplied by when is odd, then replacing by allows the tesseral harmonics to be expressed as products of factors linear in , , and multiplied by one of 1, , , , , , , and (Whittaker and Watson 1990, p. 536).

Any linear combination of real sphericalharmonicsfor fixed whose sum is not premultiplied by a factor (Whittaker and Watson 1990, p. 392).

The sum of the absolute squares of the spherical harmonics over all values of is(1)The double sum over and is given by(2)(3)where is the delta function.

A formula also known as the Legendre addition theorem which is derived by finding Green's functions for the spherical harmonic expansion and equating them to the generating function for Legendre polynomials. When is defined by(1)The Legendre polynomial of argument is given by(2)(3)(4)Another version of the formula can be given as(5)(O. Marichev, pers. comm., Jan. 15, 2008).

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..

A spherical harmonic ofthe formor

The first solution to Lamé's differential equation, denoted for , ..., . They are also called Lamé functions. The product of two ellipsoidal harmonics of the first kind is a spherical harmonic. Whittaker and Watson (1990, pp. 536-537) write(1)(2)and give various types of ellipsoidal harmonics and their highest degree terms as 1. 2. 3. 4. .A Lamé function of degree may be expressed as(3)where or 1/2, are real and unequal to each other and to , , and , and(4)Byerly (1959) uses the recurrence relations to explicitly compute some ellipsoidal harmonics, which he denoted by , , , and ,(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)

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