Angular eigenvalue

4.1 Angular eigenvalue

The solutions of the angular equation (15

) that reduce to the spin-weighted spherical harmonics in the limit $aω → 0$ are called the spin-weighted spheroidal harmonics. They are the eigenfunctions of Equation (15

), with $λ$ being the eigenvalues. The eigenvalues $λ$ are necessary for discussions of the radial Teukolsky equation. For general spin weight $s$ , the spin weighted spheroidal harmonics obey

In the post-Newtonian expansion, the parameter $aω$ is assumed to be small. Then, it is straightforward to obtain a spheroidal harmonic $S s ℓm$ of spin-weight $s$ and its eigenvalue $λ$ perturbatively by the standard method [86 , 101 , 94 ].

It is also possible to obtain the spheroidal harmonics by expansion in terms of the Jacobi functions [35 ]. In this method, if we calculate numerically, we can obtain them and their eigenvalues for an arbitrary value of $aω$ .

Here we only show an analytic formula for the eigenvalue $λ$ accurate to $2 𝒪 ((aω) )$ , which is needed for the calculation of the radial functions. It is given by

where

λ0 = ℓ(ℓ + 1) − s(s + 1), ( 2 ) λ1 = − 2m 1 + ℓ(ℓs+1) , (111 ) λ2 = H (ℓ + 1) − H (ℓ),

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