Matching of horizon and outer solutions

4.4 Matching of horizon and outer solutions

Now, we match the two types of solutions $R ν0$ and $R νC$ . Note that both of them are convergent in a very large region of $r$ , namely for $𝜖κ < ˆz < ∞$ . We see that both solutions behave as $ˆzν$ multiplied by a single-valued function of $ˆz$ for large $|ˆz|$ . Thus, the analytic properties of $R ν 0$ and $R ν C$ are the same, which implies that these two are identical up to a constant multiple. Therefore, we set

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

( ˆz ) −s−i𝜖+ ∑∞ ∞∑ R ν0 = ei𝜖κe−iˆz(𝜖κ)−ν−i𝜖+ ˆzν+i𝜖+ ---− 1 Cn,jˆzn− j 𝜖κ n=− ∞ j=0 ( ) −s−i𝜖+ ∑∞ ∞∑ = ei𝜖κe−iˆz(𝜖κ)−ν−i𝜖+ ˆzν+i𝜖+ ˆz--− 1 C zˆk, (161 ) 𝜖κ n,n− k k=− ∞ n=k ( )− s− i𝜖 ∞ ∞ ν −iˆz ν −s−i𝜖+ ν+i𝜖+ -ˆz- + ∑ ∑ n+j R C = e 2 (𝜖κ) zˆ 𝜖κ − 1 Dn,jˆz n=−∞ j=0 ( )− s− i𝜖+ ∑∞ ∑k = e−iˆz2ν(𝜖κ)−s−i𝜖+zˆν+i𝜖+ -ˆz-− 1 Dn,k−nˆzk, (162 ) 𝜖κ k=−∞ n=− ∞

where

----Γ-(1 −-s-−-2i𝜖+)-Γ (2n-+-2ν-+-1)-- Cn,j = Γ (n + ν + 1 − iτ )Γ (n + ν + 1 − s − i𝜖) × (−-n-−-ν-−-iτ)j(− n-−-ν-−-s-−-i𝜖)j(𝜖κ)−n+jfn, (163 ) (− 2n − 2ν)j(j!) Γ (n + ν + 1 − s + i𝜖)(ν + 1 + s − i𝜖)n Dn,j = (− 1)n(2i)n+j -------------------------------------- Γ (2n + 2ν + 2 ) (ν + 1 − s + i𝜖)n (n-+-ν-+-1-−-s-+-i𝜖)j- × (2n + 2ν + 2)(j!) fn. (164 ) j

Then, by comparing each integer power of $ˆz$ in the summation, in the region $𝜖κ ≪ ˆz < ∞$ , and using the formula $Γ (z)Γ (1 − z)$ $= π∕sin πz$ , we find

( r ) −1 ( ∞ ) i𝜖κ s−ν −ν ∑ ∑ K ν = e (𝜖κ) 2 Dn,r−n Cn,n−r n= −∞ n=r ei𝜖κ(2 𝜖κ)s− ν−r2−sir Γ (1 − s − 2i𝜖+)Γ (r + 2 ν + 2) = ----------------------------------------------------------- Γ (r + ν + 1 − s + i𝜖) Γ (r + ν + 1 + iτ )Γ (r + ν + 1 + s + i𝜖) ( ∞∑ ) × (− 1)nΓ (n-+-r-+-2ν-+-1)-Γ (n-+-ν-+-1-+-s +-i𝜖)-Γ (n-+-ν-+-1-+-iτ)fν n=r (n − r)! Γ (n + ν + 1 − s − i𝜖) Γ (n + ν + 1 − iτ) n ( ) −1 ∑ r (− 1)n (ν + 1 + s − i𝜖)n ν × --------------------------------------fn , (165 ) n=−∞ (r − n)!(r + 2ν + 2)n(ν + 1 − s + i𝜖)n

where $r$ can be any integer, and the factor $K ν$ 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 $𝜖$ .

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

which converges at $r > r+$ , including $r = ∞$ . 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 $r → ∞$ and $r → r +$ , they are found to be¹

( 𝜖κ )2s 2lnκ ∑∞ Btrans = --- eiκ𝜖+(1+ 1+κ ) fνn, (167 ) ω n=− ∞ ( ) Binc = ω− 1 K ν − ie−iπνsinπ-(ν −-s +-i𝜖)K −ν−1 A ν+e− i(𝜖ln𝜖− 1−κ2 𝜖), (168 ) sinπ (ν + s − i𝜖) ref − 1− 2s( iπν ) ν i(𝜖ln𝜖− 1−2κ𝜖) B = ω K ν + ie K −ν−1 A − e . (169 )

Update

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