We present a method of post-Newtonian expansion to solve the homogeneous
Regge-Wheeler equation which describes gravitational waves on the Schwarzschild
spacetime. The advantage of our method is that it allows a systematic iterative
analysis of the solution. Then we obtain the Regge-Wheeler function which is
purely ingoing at the horizon in closed analytic form, with accuracy required
to determine the gravitational wave luminosity to (post)4-Newtonian order
(i.e., order v8 beyond Newtonian) from a particle orbiting around a
Schwarzschild black hole. Our result, valid in the small-mass limit of one
body, gives an important guideline for the study of coalescing compact
binaries. In particular, it provides basic formulas to analytically calculate
detailed waveforms and luminosity, including the tail terms to
(post)3-Newtonian order, which should be reproduced in any other
post-Newtonian calculations.Comment: 31 pages, KUNS 124
