

3.3 Outer solution; 
We now solve Equation (70) in the limit
, i.e., by
applying the post-Minkowski expansion to it. In this section, we
consider the solution to
. Then we match the solution to the horizon solution
given by Equation (80) at
.
By setting
each
is
found to satisfy
Equation (84) is an inhomogeneous spherical Bessel
equation. It is the simplicity of this equation that motivated the
introduction of the auxiliary function
[49
].
The zeroth-order solution
satisfies the
homogeneous spherical Bessel equation, and must be a linear
combination of the spherical Bessel functions of the first and
second kinds,
and
. Here, we demand the compatibility with the horizon
solution (80). Since
and
,
does not match with the horizon
solution at the leading order of
. Therefore, we have
The constant
represents the overall normalization of the solution. Since it can
be chosen arbitrarily, we set
below.
The procedure to obtain
was
described in detail in [49
]. Using the Green function
, Equation (84) may be put into an indefinite integral
form,
The calculation is tedious but straightforward. All the necessary
formulae to obtain
for
are given in the Appendix of [49
] or Appendix D
of [34
]. Using those formulae, for
we have
Here,
and
are functions
defined as follows. The function
is given by
where
and
. The function
is defined by
. It is a polynomial in
inverse powers of
given by
Here, we again perform the matching with the
horizon solution (80). It should be noted that
, given by
Equation (87), is regular in the limit
except for
the term
. By examining the asymptotic behavior of
Equation (87) at
, we find
, i.e.,
the solution is regular at
. As for
, it only contributes to the renormalization of
. Hence, we set
and the
transmission amplitude
is determined to
as
It may be noted that this explicit expression for
is
unnecessary for the evaluation of gravitational waves at infinity.
It is relevant only for the evaluation of the black hole
absorption.

