Outer solution as a series of Coulomb wave functions

4.3 Outer solution as a series of Coulomb wave functions

The solution as a series of hypergeometric functions discussed in Section 4.2 is convergent at any finite value of $r$ . However, it does not converge at infinity, and hence the asymptotic amplitudes, $Binc$ and $Bref$ , cannot be determined from it. To determine the asymptotic amplitudes, it is necessary to construct a solution that is valid at infinity and to match the two solutions in a region where both solutions converge. The solution convergent at infinity was obtained by Leaver as a series of Coulomb wave functions [64

]. In this section, we review Leaver’s solution based on [69

In this section again, by noting the symmetry $¯ R ℓm ω = R ℓ− m−ω$ , we assume $ω > 0$ without loss of generality.

First, we define a variable $ˆz = ω(r − r− ) = 𝜖κ(1 − x)$ . Let us denote a Teukolsky function by $RC$ . We introduce a function $f(ˆz)$ by

Then the Teukolsky equation becomes

ˆz2f′′ + [ˆz2 + (2 𝜖 + 2is)ˆz − λ − s(s + 1)]f = 𝜖κˆz(f′′ + f) + 𝜖κ(s − 1 + 2i𝜖)f ′ 𝜖 − -[κ − i(𝜖 − mq )](s − 1 + i𝜖)f ˆz +[− 2𝜖2 + 𝜖mq + κ (𝜖2 + i𝜖s)]f. (140 )

We see that the right-hand side is explicitly of $𝒪 (𝜖)$ and the left-hand side is in the form of the Coulomb wave equation. Therefore, in the limit $𝜖 → 0$ , we obtain a solution

where $FL(η,zˆ)$ is a Coulomb wave function given by

and $Φ$ is the regular confluent hypergeometric function (see [1 ], Section 13) which is regular at $ˆz = 0$ .

In the same spirit as in Section 4.2, we introduce the renormalized angular momentum $ν$ . That is, we add $[λ + s(s + 1) − ν(ν + 1)]f (ˆz)$ to both sides of Equation (140 ) to rewrite it as

ˆz2f′′ + [ˆz2 + (2𝜖 + 2is)ˆz − ν(ν + 1)]f = 𝜖κˆz(f′′ + f) + 𝜖κ(s − 1 + 2i𝜖+)f′ 𝜖 − -[κ − i(𝜖 − mq )](s − 1 + i𝜖)f ˆz 2 2 + [− ν(ν + 1) + λ + s(s + 1) − 2𝜖 + 𝜖mq + κ(𝜖 + i𝜖s)]f. (143 )

We denote the formal solution specified by the index $ν$ by $fν(ˆz)$ , and expand it in terms of the Coulomb wave functions as

where $(a)n = Γ (a + n)∕Γ (a)$ . Then, using the recurrence relations among $Fn+ ν$ ,

1 (n + ν + 1 + s − i𝜖) zFn+ ν = (n-+-ν-+-1)(2n-+-2-ν +-1)Fn+ν+1 + ------is-+-𝜖------F (n + ν)(n + ν + 1) n+ν (n + ν − s + i𝜖) + ---------------------Fn+ν−1, (145 ) (n + ν)(2n + 2ν + 1 ) ′ (n-+-ν)(n-+-ν-+-1-+-s-−-i𝜖) F n+ν = − (n + ν + 1)(2n + 2ν + 1 ) Fn+ν+1 + ------is-+-𝜖------F (n + ν)(n + ν + 1) n+ν (n + ν + 1)(n + ν − s + i𝜖) + ---------------------------Fn+ν− 1, (146 ) (n + ν)(2n + 2ν + 1 )

we can derive the recurrence relation among $bn$ . The result turns out to be identical to the one given by Equation (123

) for $a n$ . We mention that the extra factor $(ν + 1 + s − i𝜖) ∕(ν + 1 − s + i𝜖) n n$ in Equation (144

) is introduced to make the recurrence relation exactly identical to Equation (123

The fact that we have the same recurrence relation as Equation (123 ) implies that if we choose the parameter $ν$ in Equation (144 ) to be the same as the one given by a solution of Equation (133 ) or (136 ), the sequence ${f ν} n$ is also the solution for ${b } n$ , which is minimal for both $n → ± ∞$ . Let us set

By choosing $ν$ as stated above, we have the asymptotic value for the ratio of two successive terms of $gνn$ as

Using an asymptotic property of the Coulomb wave functions, we have

We thus find that the series (144

) converges at $ˆz > 𝜖κ$ or equivalently $r > r+$ .

The fact that we can use the same $ν$ as in the case of hypergeometric functions to obtain the convergence of the series of the Coulomb wave functions is crucial to match the horizon and outer solutions.

Here, we note an analytic property of the confluent hypergeometric function (see [34 ], Page 259),

where $Ψ$ is the irregular confluent hypergeometric function, and $Im (x) > 0$ is assumed. Using this with the identities

a = n + ν + 1 − s + i𝜖, c = 2(n + ν + 1), (151 ) x = 2iˆz,

we can rewrite $ν R C$ (for $ω > 0$ ) as

where

ν ν −π𝜖 iπ(ν+1−s)Γ (ν + 1 − s + i𝜖) −iˆz ν+i𝜖+ −s−i𝜖+ R + = 2 e e Γ (ν-+-1-+-s −-i𝜖)-e ˆz (ˆz − 𝜖κ) ∑∞ × inf ν(2ˆz)nΨ (n + ν + 1 − s + i𝜖,2n + 2ν + 2; 2izˆ), n n=− ∞ (153 ) R ν− = 2 νe−π𝜖e−iπ(ν+1+s)eiˆzˆzν+i𝜖+(ˆz − 𝜖κ)−s−i𝜖+ ∞ ∑ n(ν-+-1-+-s −-i𝜖)n- ν n × i (ν + 1 − s + i𝜖)n fn(2ˆz) n= −∞ × Ψ (n + ν + 1 + s − i𝜖,2n + 2ν + 2;− 2izˆ).

By noting an asymptotic behavior of $Ψ (a,c;x)$ at large $|x|$ ,

we find

where

+∑ ∞ A ν+ = e− (π∕2)𝜖e(π∕2)i(ν+1− s)2−1+s− i𝜖Γ (ν-+-1-−-s-+-i𝜖) fνn, (157 ) Γ (ν + 1 + s − i𝜖)n=− ∞ +∞ ν −1−s+i𝜖 −(π∕2)i(ν+1+s) −(π∕2)𝜖 ∑ n (ν-+-1-+-s-−-i𝜖)n-ν A − = 2 e e (− 1) (ν + 1 − s + i𝜖)nfn. (158 ) n= −∞

We can see that the functions $R ν +$ and $R ν −$ are incoming-wave and outgoing wave solutions at infinity, respectively. In particular, we have the upgoing solution, defined for $s = − 2$ by the asymptotic behavior (20

), expressed in terms of a series of Coulomb wave functions as