In the region , we may expand both solutions in powers of except for analytically non-trivial factors. We have
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 be1Update
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
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