New Numerical Methods to Evaluate Homogeneous Solutions of the Teukolsky Equation

We discuss a numerical method to compute the homogeneous solutions of theTeukolsky equation which is the basic equation of the black hole perturbationmethod. We use the formalism developed by Mano, Suzuki and Takasugi, in whichthe homogeneous solutions of the radial Teukolsky equation are expressed interms of two kinds of series of special functions, and the formulas for theasymptotic amplitudes are derived explicitly.Although the application of thismethod was previously limited to the analytical evaluation of the homogeneoussolutions, we find that it is also useful for numerical computation. We alsofind that so-called "renormalized angular momentum parameter", $\nu$, can befound only in the limited region of $\omega$ for each $l,m$ if we assume $\nu$is real (here, $\omega$ is the angular frequency, and $l$ and $m$ are degreeand order of the spin-weighted spheroidal harmonics respectively). We alsocompute the flux of the gravitational waves induced by a compact star in acircular orbit on the equatorial plane around a rotating black hole. We findthat the relative error of the energy flux is about $10^{-14}$ which is muchsmaller than the one obtained by usual numerical integration methods.Comment: 36 pages,7 figure