We review the analytic methods used to perform the post-Newtonian expansion of gravitational waves induced by a particle orbiting a massive, compact body, based on black hole perturbation theory. There exist two different methods of performing the post-Newtonian expansion. Both are based on the Teukolsky equation. In one method, the Teukolsky equation is transformed into a Regge–Wheeler type equation that reduces to the standard Klein–Gordon equation in the flat-space limit, while in the other method (which was introduced by Mano, Suzuki, and Takasugi relatively recently), the Teukolsky equation is used directly in its original form. The former’s advantage is that it is intuitively easy to understand how various curved space effects come into play. However, it becomes increasingly complicated when one goes to higher and higher post-Newtonian orders. In contrast, the latter’s advantage is that a systematic calculation to higher post-Newtonian orders can be implemented relatively easily, but otherwise, it is so mathematical that it is hard to understand the interplay of higher order terms. In this paper, we review both methods so that their pros and cons may be seen clearly. We also review some results of calculations of gravitational radiation emitted by a particle orbiting a black hole.
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