

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 [33
], who found analytic expressions
of the solutions of the homogeneous Teukolsky equation. In this
method, the exact solutions of the radial Teukolsky equation (15) 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 [48
]. 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.

