H_5

This is the Title PhL 1.11.15 do not edit, has been installed on 12/18/15 H.5 Limits of toroidal functions (and combinations) as n→∞ 1 H.6 Limits of toroidal functions (and combinations) as z→∞ 3 H.7 Limits of toroidal functions as z→1 5 H.8 Alternate series for T(z) near z = 1 7 H.5 Limits of toroidal functions (and combinations) as n→∞ Results below assume ξ ≥ 0 and ξ0 ≥ 0. From AS2010 p 366 and p 354 we find that, for large ν, (H.5.1) Our Q function of interest is the unbolded one. Therefore, Pν(chξ) → I0[(ν+1/2)ξ] Qν(chξ) → K0[(ν+1/2)ξ] ν→∞ (H.5.2) so for our toroidal functions. Pn-1/2(chξ) → I0(nξ) Qn-1/2(chξ) → K0(nξ) . n→∞ (H.5.3) Here I0 and K0 are modified Bessel functions. From the same source the large z behaviors of I0 and Ko are given by, with . (H.5.4) Therefore, I0(z) → ez K0(z) → e-z z→∞ (H.5.5) so I0(nξ) → enξ K0(nξ) → e-nξ . n→∞ (H.5.6) Putting the pieces together, we find that Pn-1/2(chξ0) → enξ = enξ Qn-1/2(chξ0) → e-nξ = e-nξ . n→∞ (H.5.7) so Pn-1/2(chξ0) diverges exponentially as n→ ∞, whereas Qn-1/2(chξ0) converges exponentially. We are interested several special combinations of functions. First, Pn-1/2(chξ0)Qn-1/2(chξ0) → n→∞ (H.5.8) which is only mildly convergent as n→ ∞. Next, → π e-2nξ n→∞ (H.5.9) which is exponentially convergent. Finally, Pn-1/2(chξ) → e-n(2ξ-ξ) n→∞ (H.5.10) which is exponentially convergent for ξ < 2ξ0. H.6 Limits of toroidal functions (and combinations) as z→∞ (a) The cases n = 1,2,3.... We start with this x→ ∞ form from AS2010, (H.6.1) Setting μ = 0 we find that Pν(x) → (1/)[Γ(ν+1/2)/Γ(ν+1)] (2x)ν ν > -1/2 x → ∞ (H.6.2) Setting ν = n-1/2 for n = 0,1,2... one finds, Pn-1/2(x) → (1/)[Γ(n)/Γ(n+1/2)] (2x)n-1/2 n > 0 x → ∞ . (H.6.3) For the Q functions one has instead, (H.6.4) Then Qν(x) → [Γ(ν+1)/Γ(ν+3/2)] (2x)-ν-1 . (H.6.5) Setting ν = n-1/2 for n = 0,1,2... Qn-1/2(x) → [Γ(n+1/2)/Γ(n+1)] (2x)-n-1/2 n = 0,1,2... x→∞ . (H.6.6) A ratio of interest is this, → = π (2x)-2n n = 1,2.... x→∞ (H.6.7) (b) The case n = 0 For Q-1/2(x) we use (H.6.6) with n = 0 above to get Q-1/2(x) → [Γ(1/2)/Γ(1)] (2x)-1/2 = (π/) x-1/2 . (H.6.8) The P-1/2(x) function requires much more work, it is a special case. We start with AS2010 , (H.6.9) which relates P-1/2(x) to the complete elliptical integral K(z). As ξ→∞, tanh(ξ/2) → 1 so we need the behavior of K(z) for z near 1. GR7 8.113.3 gives this expansion of K(k) near k = 1 (small k'), where (H.6.10) Keeping only the leading term, K(k) ≈ ln(4/k') = ln(4/) . (H.6.11) Now if k = th(ξ/2), then k' = = sech(ξ/2) so, K( th(ξ/2) ) ≈ ln(4 ch(ξ/2)) = (1/2) ln(16 ch2(ξ/2)] = (1/2)ln[ 8 (chξ +1)] . (H.6.12) Therefore P-1/2(chξ) = K(th(ξ/2)) ≈ (1/2) ln[ 8 (1+chξ)] ≈ ln(8chξ) large ξ (H.6.13) which is more simply stated as P-1/2(x) ≈ x-1/2 ln(8x) . large x (H.6.14) Notice from (H.6.3) that the other Pn-1/2(x) have fast power decay (2x)n-1/2, but P-1/2(x) has a very slow We then compute our special n = 0 case Q/P ratio from (H.6.8) and (H.6.14) to be → = (π2/2) 1/ln(8x) x→∞ (H.6.15) One could argue that ln(8x) = ln(8) + ln(x) ≈ ln(x) for large x, but that would require very large x indeed. For example, if ln(x) > 20 ln(8), then x > 820 ~ 1017. So we keep the 8. H.7 Limits of toroidal functions as z→1 From AS2010 as x → 0 from above we have (H.7.1) Thus we conclude that Pν(x) → 1 x → 1 (H.7.2) which is a well known result. The Q function is more complicated : (H.7.3) The first term shows logarithmic divergence as x→ 1. The second term is a constant where γ is "Euler's constant" (0.557) and ψ(x) is the psi function (the digamma function), ψ(x) = ∂xlnΓ(x) = . (H.7.4) Setting ν = n-1/2 we find that Pn-1/2(x) → 1 x → 1 Qn-1/2(x) → - ln(x-1) + x → 1 (H.7.5) To obtain the behavior of Pn-1/2(x) near x = 1, we use (7.4.1) P(ν,ξ) ≡ Pν(chξ) = (chξ)ν F(-ν/2, 1/2-ν/2; 1; th2ξ) (7.4.1) where x = chξ. For small ξ we find, using F(a,b;c;z) ≈ 1 + (ab/c) z + O(z2) and thξ ≈ ξ, P(ν,ξ) ≈ (1 + ξ2/2)ν [ 1 + (-ν/2)(1/2-ν/2) ξ2 ] = (1 + νξ2/2)( 1 + (ν/2)(ν/2-1/2) ξ2 = 1 + [(1/2)ν + (1/4)ν(ν-1)]ξ2 = 1 + (1/4)[2ν +ν(ν-1) ]ξ2 = 1 + (1/4)[ν2+ν]ξ2 = 1 + (1/4)ν(ν+1)ξ2 . (H.7.6) Then Pn-1/2(chξ) ≈ 1 + (1/4)(n-1/2)(n+1/2)ξ2 ≈ 1 + (1/4) [n2 - 1/4]ξ2 ξ→0 (H.7.7) Setting z = chξ ≈ 1+ξ2/2 we have ξ2 ≈ 2(z-1) so then Pn-1/2(z) ≈ 1 + (1/2) [n2 - 1/4](z-1) (H.7.8) Except for n = 0 the correction term is positive, therefore Pn-1/2(z) ≥ 1 as z → 1+ for n = 1,2...∞ (H.7.9) Application: Consider T(z) ≡ Σn=0∞ εn [Qn-1/2(z) / Pn-1/2(z)] = * 2 * [Q-1/2(z) / P-1/2(z)] + Σn=1∞ εn [Qn-1/2(z) / Pn-1/2(z)] . For z very close to 1 we install P-1/2(z) = 1 to get T(z) ≈ * 2 * Q-1/2(z) + Σn=1∞ εn [Qn-1/2(z) / Pn-1/2(z)] . But for n > 0 we know from (H.7.9) that Pn-1/2(z) ≥ 1, so [1/ Pn-1/2(z)] ≤ 1. Therefore, T(z) ≤ * 2 * Q-1/2(z) + Σn=1∞ εn [Qn-1/2(z) ] or T(1) ≤ limz→1+ ( Σn=0∞ εn [Qn-1/2(z)] ) or T(1) ≤ limz→1+ ( Σn=0∞ εn [Qn-1/2(z)] . (H.7.10) In (B.2.17) it is shown that Σn=0∞ εn Qn-1/2(z) = (π/) (B.2.17) and therefore we may conclude that T(1) ≤ π (H.7.11) H.8 Alternate series for T(z) near z = 1 From AS2010 the Wronskian of the P and Q solutions of the Legendre equation is given by. (H.8.1) so that W{Pν(z),Qν(z)} = 1/(1-z2) = Pν(z) Qν'(z) - Pν'(z) Qν(z) . (H.8.2) Now consider the series T(z) ≡ Σn=0∞ εn [Qn-1/2(z) / Pn-1/2(z)] = f(z) f(z) ≡ Σn=0∞ εn [Qn-1/2(z) / Pn-1/2(z)] . (H.8.3) Then f '(z) = Σn=0∞ εn [Pn-1/2(z) Q'n-1/2(z) - Qn-1/2(z) P'n-1/2(z)] / [ Pn-1/2(z)]2 = Σn=0∞ εn [ 1/(1-z2) ] / [ Pn-1/2(z)]2 // (H.8.2) = - (z2-1)-1 Σn=0∞ εn 1/ [ Pn-1/2(z)]2 . (H.8.4) Then T'(z) = ()' f(z) + f '(z) = (z /) f(z) + ( - (z2-1)-1 Σn=0∞ εn 1/ [ Pn-1/2(z)]2 ) = (1/) [ z f(z) - Σn=0∞ εn 1/ [ Pn-1/2(z)]2 ] (H.8.5) so that T'(z) = z f(z) - Σn=0∞ εn 1/ [ Pn-1/2(z)]2 . (H.8.6) In the limit z→1, assuming T'(z) is not infinite, this says 0 = f(z) - Σn=0∞ εn 1/ [ Pn-1/2(z)]2 or f(z) = Σn=0∞ εn 1/ [ Pn-1/2(z)]2 . // for z very close to 1 (H.8.7) Then from (H.8.3), T(z) = f(z) = Σn=0∞ εn for z very close to 1 . (H.8.8)