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 . 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 byUpdate
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