Zonal spherical harmonic

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 by

where P_ℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by ${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}$, 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 ${\displaystyle Z^{(\ell )}(\theta ,\phi ).}$

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