the hermitian form Specifically, is a spin-weight function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism. The operator The spin weight bundles are equipped with a differential operator (eth). This operator is essentially the Dolbeault operator, after suitable identifications have been made, Thus for , defines a function of spin-weight . Spin-weighted harmonics Just as conventional spherical harmonics are the eigenfunctions of the Laplace-Beltrami operator on the sphere, the spin-weight harmonics are the eigensections for the Laplace-Beltrami operator acting on the bundles of spin-weight functions. Representation as functions The spin-weighted harmonics can be represented as functions on a sphere once a point on the sphere has been selected to serve as the North pole. By definition, a function with spin weight transforms under rotation about the pole via Working in standard spherical coordinates, we can define a particular operator acting on a function as: This gives us another function of and . (The operator is effectively a covariant derivative operator in the sphere.) An important property of the new function is that if had spin weight , has spin weight . Thus, the operator raises the spin weight of a function by 1. Similarly, we can define an operator which will lower the spin weight of a function by 1: The spin-weighted spherical harmonics are then defined in terms of the usual spherical harmonics as: The functions then have the property of transforming with spin weight . Other important properties include the following: Orthogonality and completeness The harmonics are orthogonal over the entire sphere: and satisfy the completeness relation Calculating These harmonics can be explicitly calculated by several methods. The obvious recursion relation results from repeatedly applying the raising or lowering operators. Formulae for direct calculation were derived by . Note that their formulae use an old choice for the Condon–Shortley phase. The convention chosen below is in agreement with Mathematica, for instance. The more useful of the Goldberg, et al., formulae is the following: A Mathematica notebook using this formula to calculate arbitrary spin-weighted spherical harmonics can be found here. With the phase convention here: First few spin-weighted spherical harmonics Analytic expressions for the first few orthonormalized spin-weighted spherical harmonics: Spin-weight , degree Relation to Wigner rotation matrices This relation allows the spin harmonics to be calculated using recursion relations for the -matrices. Triple integral The triple integral in the case that is