4.1 Angular eigenvalue

4.1 Angular eigenvalue

The solutions of the angular equation (16

) 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 (16

), 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 [46 , 58 , 52 ].

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)