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 [64Jump To The Next Citation Point]. In this section, we review Leaver’s solution based on [69Jump To The Next Citation Point].

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

− 1− s( 𝜖κ )−s−i(𝜖+τ)∕2 RC = ˆz 1 − -ˆz- f(ˆz). (139 )
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
f (zˆ) = F ℓ(− is − 𝜖, ˆz), (141 )
where FL(η,zˆ) is a Coulomb wave function given by
F (η, ˆz) = e− iˆz2LˆzL+1 Γ (L-+-1 −-iη)Φ (L + 1 − iη,2L + 2;2iˆz), (142 ) L Γ (2L + 2 )
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 (140View Equation) 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
∞∑ f = (− i)n(ν-+-1-+-s-−-i𝜖)n-b F (− is − 𝜖,zˆ), (144 ) ν (ν + 1 − s + i𝜖)n n n+ν n=−∞
where (a)n = Γ (a + n)âˆ•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 (123View Equation) for a n. We mention that the extra factor (ν + 1 + s − i𝜖) ∕(ν + 1 − s + i𝜖) n n in Equation (144View Equation) is introduced to make the recurrence relation exactly identical to Equation (123View Equation).

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

(ν + 1 + s − i𝜖)n gνn = (− i)n ----------------fnν. (147 ) (ν + 1 − s + i𝜖)n
By choosing ν as stated above, we have the asymptotic value for the ratio of two successive terms of gνn as
gν gν 𝜖κ lim n -νn--= lim n -νn--= ---. (148 ) n→∞ gn− 1 n→− ∞ gn+1 2
Using an asymptotic property of the Coulomb wave functions, we have
gνFn+ ν(z ) gνFn+ ν(z) 𝜖κ lim -ν-n----------= lim -ν-n---------- = ---. (149 ) n→∞ gn−1Fn+ ν− 1(z) n→− ∞ gn−1Fn+ ν+1(z) z
We thus find that the series (144View Equation) 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),

Γ (c) Γ (c) Φ(a,c;x ) = --------eiaπΨ(a,c;x ) +-----eiπ(a−c)exΨ (c − a,c;− x), (150 ) Γ (c − a) Γ (a)
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
ν ν ν R C = R + + R − , (152 )
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|,
− a Ψ(a, c; x) → x as |x| → ∞, (154 )
we find
R ν = A νz −1e−i(z+𝜖lnz), (155 ) + + R ν− = A ν− z −1−2sei(z+𝜖lnz), (156 )
where
+∑ ∞ A ν+ = e− (π∕2)𝜖e(π∕2)i(ν+1− s)2−1+s− iðœ–Γ (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 (20View Equation), expressed in terms of a series of Coulomb wave functions as
Rup = R ν− . (159 )

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