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- Zonal_spherical_harmonics abstract "In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in spherical coordinates bywhere Pℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by , where x is a point on the sphere representing the fixed axis, and y is the variable of the function. This can be obtained by rotation of the basic zonal harmonic In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define to be the dual representation of the linear functionalin the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:for all Y ∈ Hℓ. The integral is taken with respect to the invariant probability measure.".
- Zonal_spherical_harmonics wikiPageID "22952896".
- Zonal_spherical_harmonics wikiPageRevisionID "511522109".
- Zonal_spherical_harmonics hasPhotoCollection Zonal_spherical_harmonics.
- Zonal_spherical_harmonics subject Category:Rotational_symmetry.
- Zonal_spherical_harmonics subject Category:Special_hypergeometric_functions.
- Zonal_spherical_harmonics type Abstraction100002137.
- Zonal_spherical_harmonics type Function113783816.
- Zonal_spherical_harmonics type MathematicalRelation113783581.
- Zonal_spherical_harmonics type Relation100031921.
- Zonal_spherical_harmonics type SpecialHypergeometricFunctions.
- Zonal_spherical_harmonics comment "In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis.".
- Zonal_spherical_harmonics label "Zonal spherical harmonics".
- Zonal_spherical_harmonics sameAs m.063_wxj.
- Zonal_spherical_harmonics sameAs Q8073864.
- Zonal_spherical_harmonics sameAs Q8073864.
- Zonal_spherical_harmonics sameAs Zonal_spherical_harmonics.
- Zonal_spherical_harmonics wasDerivedFrom Zonal_spherical_harmonics?oldid=511522109.
- Zonal_spherical_harmonics isPrimaryTopicOf Zonal_spherical_harmonics.