4 Analytic Solutions of the Homogeneous Teukolsky Equation by Means of the Series Expansion of
Special Functions
In this section, we review a method developed by Mano, Suzuki, and Takasugi [68
], who found analytic
expressions of the solutions of the homogeneous Teukolsky equation. In this method, the exact solutions of
the radial Teukolsky equation (14) are expressed in two kinds of series expansions. One is given by a series
of hypergeometric functions and the other by a series of the Coulomb wave functions. The former is
convergent at horizon and the latter at infinity. The matching of these two solutions is done exactly in the
overlapping region of convergence. They also found that the series expansions are naturally related to the
low frequency expansion. Properties of the analytic solutions were studied in detail in [69]. Thus, the
formalism is quite powerful when dealing with the post-Newtonian expansion, especially at higher
orders.
In many cases, when we study the perturbation of a Kerr black hole, it is more convenient to use the
Sasaki–Nakamura equation, since it has the form of a standard wave equation, similar to the
Regge–Wheeler equation. However, it is not quite suited for investigating analytic properties of the solution
near the horizon. In contrast, the Mano–Suzuki–Takasugi (MST) formalism allows us to investigate analytic
properties of the solution near the horizon systematically. Hence, it can be used to compute the
higher order post-Newtonian terms of the gravitational waves absorbed into a rotating black
hole.
We also note that this method is the only existing method that can be used to calculate the
gravitational waves emitted to infinity to an arbitrarily high post-Newtonian order in principle.
) are expressed in two kinds of series expansions. One is given by a series
of hypergeometric functions and the other by a series of the Coulomb wave functions. The former is
convergent at horizon and the latter at infinity. The matching of these two solutions is done exactly in the
overlapping region of convergence. They also found that the series expansions are naturally related to the
low frequency expansion. Properties of the analytic solutions were studied in detail in [69
