4.1 Angular eigenvalue

The solutions of the angular equation (15View Equation) 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 (15View Equation), 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
{ [ ] } -1---d-- -d- 2 2 2 (m--+-s-cos𝜃)2- sin 𝜃 d𝜃 sin 𝜃d𝜃 − a ω sin 𝜃 − sin2 𝜃 − 2aωs cos𝜃 + s + 2ma ω + λ sSℓm = 0.(109 )

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, 101Jump To The Next Citation Point, 94Jump To The Next Citation Point].

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

λ = λ0 + aω λ1 + a2ω2λ2 + 𝒪 ((aω)3), (110 )
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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2(ℓ2 − m2 )(ℓ2 − s2)2 H (ℓ) = ----------3---------. (112 ) (2ℓ − 1)ℓ (2 ℓ + 1)

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