In the region , we may expand both
solutions in powers of
except for analytically non-trivial factors. We
have
Then, by comparing each integer power of in the summation, in
the region
, and using the formula
, we find
We thus have two expressions for the ingoing wave
function . One
is given by Equation (116
), with
expressed in terms of a series of
hypergeometric functions as given by Equation (120
) (a series which converges everywhere
except at
).
The other is expressed in terms of a series of Coulomb wave
functions given by
Now we can obtain analytic expressions for the
asymptotic amplitudes of ,
,
, and
. By investigating the asymptotic behaviors of the
solution at
and
, they are found to be
Incidentally, since we have the upgoing solution
in the outer region (159), it is straightforward to obtain the
asymptotic outgoing amplitude at infinity
from
Equation (153
). We find