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4.1 Angular eigenvalue

The solutions of the angular equation (16View Equation) that reduce to the spin-weighted spherical harmonics in the limit aw --> 0 are called the spin-weighted spheroidal harmonics. They are the eigenfunctions of Equation (16View Equation), with c being the eigenvalues. The eigenvalues c are necessary for discussions of the radial Teukolsky equation. For general spin weight s, the spin weighted spheroidal harmonics obey
{ [ ] 2 } -1---d-- sin h-d- - a2w2sin2h - (m--+-s-cosh)--- 2aws cosh + s + 2maw + c sSlm = 0.(109) sin h dh dh sin2 h

In the post-Newtonian expansion, the parameter aw is assumed to be small. Then, it is straightforward to obtain a spheroidal harmonic sSlm of spin-weight s and its eigenvalue c perturbatively by the standard method [4658Jump To The Next Citation Point52Jump To The Next Citation Point].

It is also possible to obtain the spheroidal harmonics by expansion in terms of the Jacobi functions [21]. In this method, if we calculate numerically, we can obtain them and their eigenvalues for an arbitrary value of aw.

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

c = c0 + awc1 + a2w2c2 + O((aw)3), (110)
where
c0 = l(l + 1) - s(s + 1), ( 2 ) c1 = -2m 1 + l(ls+1) , (111) c2 = 1 + (H(l + 1)- H(l) - 1),
with
2(l2- m2)(l2 - s2)2 H(l) = -(2l---1)l3(2l-+-1)--. (112)


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