### 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  [91].

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 [91]. 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 [91] or Appendix D of [71]. 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.