### 4.1 Angular eigenvalue

The solutions of the angular equation (15) that reduce to the spin-weighted spherical harmonics in the
limit are called the spin-weighted spheroidal harmonics. They are the eigenfunctions of
Equation (15), with being the eigenvalues. The eigenvalues are necessary for discussions of the
radial Teukolsky equation. For general spin weight , the spin weighted spheroidal harmonics obey
In the post-Newtonian expansion, the parameter is assumed to be small. Then, it is
straightforward to obtain a spheroidal harmonic of spin-weight and its eigenvalue
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 .

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

where
Update