

3.2 Horizon solution; 
In this section, we first consider the solution near the horizon,
which we call the horizon solution, based on [45
]. To do so, we assume
and treat
as a small number,
but leave the ratio
arbitrary. We change the independent variable to
and
the wave function to
Note that the horizon corresponds to
. We then have
where a prime denotes differentiation with respect to
. We look for a
solution which is regular at
.
First, we consider the lowest order solution by
setting
in
Equation (74). The boundary condition (72) requires that
at
. The solution
that satisfies the boundary condition is
Thus, the lowest order solution is a polynomial of order
in
.
Next, we consider the solution accurate to
. We neglect the
terms of
in
Equation (74). Then, the wave equation takes the form
of a hypergeometric equation,
with parameters
The two linearly independent solutions are
and
, where
is the hypergeometric function. However, only the
first solution is regular at
. Therefore, we obtain
The above solution must be matched with the
solution obtained from the post-Minkowski expansion of
Equation (70), which we call the outer solution, in a
region where both solutions are valid. It is the region where the
post-Newtonian expansion is applied, i.e., the region
. For this purpose, we rewrite
Equation (78) as (see, e.g., Equation (15.3.8) of [1
])
This naturally allows the expansion in
. It should be noted that the second term
in the square brackets of the above expression is meaningless as it
is, since the factor
diverges for integer
. So, when evaluating the second term, we
first have to extend
to a non-integer number. Then, only after expanding
it in terms of
,
we should take the limit of an integer
. One then finds that this procedure gives
rise to an additional factor of
. For
, it therefore becomes
higher in
than the first term.
Then, we obtain
where
and
is the
digamma function,
and
is the Euler constant.
As we will see below, the above solution is
accurate enough to determine the boundary condition of the outer
solution up to the 6PN order of expansion.

