4.4 Matching of horizon and outer solutions

4.4 Matching of horizon and outer solutions

Now, we match the two types of solutions $Rn0$ and $RnC$ . Note that both of them are convergent in a very large region of $r$ , namely for $ek < ^z < oo$ . We see that both solutions behave as $^zn$ multiplied by a single-valued function of $^z$ for large $|^z|$ . Thus, the analytic properties of $Rn 0$ and $Rn C$ are the same, which implies that these two are identical up to a constant multiple. Therefore, we set

In the region $ek < ^z < oo$ , we may expand both solutions in powers of $z^$ except for analytically non-trivial factors. We have

( ^z ) -s-ie+ sum oo oo sum Rn0 = eieke-i^z(ek)-n-ie+ ^zn+ie+ ---- 1 Cn,j^zn- j ek n=- oo j=0 ( ) -s-ie+ sum oo oo sum = eieke-i^z(ek)-n-ie+ ^zn+ie+ ^z--- 1 C z^k, (161) ek n,n- k k=- oo n=k ( )- s- ie oo oo n -i^z n -s-ie+ n+ie+ -^z- + sum sum n+j R C = e 2 (ek) z^ ek - 1 Dn,j^z n=- oo j=0 ( )- s- ie+ sum oo sum k = e-i^z2n(ek)-s-ie+z^n+ie+ -^z-- 1 Dn,k-n^zk, (162) ek k=- oo n=- oo

where

----G(1---s---2ie+)-G(2n-+-2n-+-1)---- Cn,j = G(n + n + 1- it )G(n + n + 1- s- ie) × (--n---n---it)j(-n---n---s---ie)j(ek)-n+jf , (163) (-2n - 2n)j(j!) n G(n + n + 1 - s + ie)(n + 1 + s- ie) Dn,j = (- 1)n(2i)n+j ------------------------------------n- G(2n + 2n + 2) (n + 1- s + ie)n (n + n + 1 - s + ie)j × --------------------fn. (164) (2n + 2n + 2)j(j!)

Then, by comparing each integer power of $z^$ in the summation, in the region $ek « ^z < oo$ , and using the formula $G(z) G(1 - z)$ $= p/ sin pz$ , we find

( r ) -1 ( oo ) iek s-n -n sum sum Kn = e (ek) 2 Dn,r-n Cn,n-r n= - oo n=r eiek(2ek)s- n-r2-sir G(1 - s - 2ie+)G(r + 2n + 2) = ----------------------------------------------------------- G(r + n + 1 - s + ie) G(r + n + 1 + it )G(r + n + 1 + s + ie) ( oo sum ) × (- 1)nG(n-+-r-+-2n-+-1)-G(n-+-n-+-1-+-s +-ie)-G(n-+-n-+-1-+-it)fn n=r (n- r)! G(n + n + 1 - s- ie) G(n + n + 1 - it) n ( ) -1 sum r (-1)n (n + 1 + s - ie)n n × --------------------------------------fn , (165) n=- oo (r - n)!(r + 2n + 2)n(n + 1 - s + ie)n

where $r$ can be any integer, and the factor $K n$ should be independent of the choice of $r$ . Although this fact is not manifest from Equation (165

), we can check it numerically, or analytically by expanding it in terms of $e$ .

We thus have two expressions for the ingoing wave function $Rin$ . One is given by Equation (116 ), with $pn in$ expressed in terms of a series of hypergeometric functions as given by Equation (120 ) (a series which converges everywhere except at $r = oo$ ). The other is expressed in terms of a series of Coulomb wave functions given by

which converges at $r > r+$ , including $r = oo$ . 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 $Rin$ , $Btrans$ , $Binc$ , and $Bref$ . By investigating the asymptotic behaviors of the solution at $r --> oo$ and $r-- > r+$ , they are found to be

(ek )2s sum oo Btrans = --- eie+lnk fnn, (167) w n=- oo ( ) Binc = w -1 Kn - ie- ipnsinp(n----s +-ie)K -n- 1 An+e -ielne, (168) sinp(n + s- ie) Bref = w -1-2s(K + ieipnK )An eielne. (169) n -n-1 -

Incidentally, since we have the upgoing solution in the outer region (159 ), it is straightforward to obtain the asymptotic outgoing amplitude at infinity $Ctrans$ from Equation (153 ). We find

trans -1- 2s ielne n C = w e A - . (170)