### 4.5 Low frequency expansion of the hypergeometric expansion

So far, we have considered exact solutions of the Teukolsky equation. Now, let us consider their low
frequency approximations and determine the value of . We solve Equation (136) with ,
with a requirement that as .
To solve Equation (174), 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 (174) 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:

- For , we have , , and .
- For , we have , , and .
- For , we have , , and .

These imply that , , and , respectively. To
summarize, we have

With these results, we can calculate the value of to , which is given by
Now one can take the limit of an integer value of . In particular, the above holds also for .
Interestingly, is found to be independent of the azimuthal eigenvalue to .
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 [68]. A calculation up to a much higher order in was performed in [98], in which
the black hole absorption of gravitational waves was calculated to beyond the lowest
order.