To solve Equation (171), we first note the following. Unless
the value of
is
such that the denominator in the expression of
or
happens to vanish,
or
happens
to vanish in the limit
, we have
,
, and
. Also, from the asymptotic
behavior of the minimal solution
as
given by Equation (134
), we have
and
for sufficiently large
. Thus, except for
exceptional cases mentioned above, the order of
in
increases as
increases. That
is, the series solution naturally gives the post-Minkowski
expansion.
First, let us consider the case of for
. It is easily
seen that
,
, and
for all
. Therefore, we have
for all
.
On the other hand, for , the order of
behaves
irregularly for certain values of
. For the moment, let us assume that
. We see
from Equations (124
) that
,
, since
. Then, Equation (171
) implies
. Using the expansion of
given by
Equation (110
), we then find
(i.e., there is no term of
in
). With this estimate
of
, we see from
Equation (128
) that
is justified if
is of order
unity or smaller.
The general behavior of the order of in
for general values of
is rather
complicated. However, if we assume
to be a non-integer and
, and
,
it is relatively easily studied. With the assumption that
, we find
there are three exceptional cases:
These imply that ,
, and
,
respectively. To summarize, we have
The post-Minkowski expansion of homogeneous
Teukolsky functions can be obtained with arbitrary accuracy by
solving Equation (123) to a desired order, and by summing up
the terms to a sufficiently large
. The first few terms of the coefficients
are explicitly
given in [33
]. A calculation up to a much
higher order in
was performed in [55
], in which the black hole
absorption of gravitational waves was calculated to
beyond the lowest
order.