Calculating
These harmonics can be explicitly calculated by several methods. The obvious recursion relation results from repeatedly applying the raising or lowering operators. Formulas for direct calculation were derived by Goldberg et al. (1967). Note that their formulas 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., formulas 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 and .
Read more about this topic: Spin-weighted Spherical Harmonics
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