4.4 Matching of horizon and outer solutions

Now, we match the two types of solutions and . Note that both of them are convergent in a very large region of , namely for . We see that both solutions behave as multiplied by a single-valued function of for large . Thus, the analytic properties of and are the same, which implies that these two are identical up to a constant multiple. Therefore, we set

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

where
Then, by comparing each integer power of in the summation, in the region , and using the formula , we find
where can be any integer, and the factor should be independent of the choice of . Although this fact is not manifest from Equation (165), we can check it numerically, or analytically by expanding it in terms of .

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

which converges at , including . Combining these two, we have a complete analytic solution for the ingoing wave function.

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