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New Numerical Methods to Evaluate Homogeneous Solutions of the Teukolsky Equation II. Solutions of the Continued Fraction Equation

We investigate the solution of the continued fraction equation by which we determine "the renormalized angular momentum parameter", $ν$, in the formalism developed by Leaver and Mano, Suzuki, and Takasugi. In this formalism, we describe the homogeneous solutions of the radial Teukolsky equation, which is the basic equation of the black hole perturbation formalism. We find that, contrary to the assumption made in previous works, the solution, $ν$, becomes complex valued as $ω$ (the angular frequency) becomes large for each $l$ and $m$ (the degree and order of the spin-weighted spheroidal harmonics). We compare the power radiated by gravitational waves from a particle in a circular orbit in the equatorial plane around a Kerr black hole in two ways, one using the Mano-Suzuki-Takasugi formalism with complex $ν$ and the other using a direct numerical integration method. We find that the two methods produce consistent results. These facts prove the validity of using complex solutions to determine the homogeneous solutions of the Teukolsky equation.