Mehler Fock delta function expansion

1 The generalized Mehler-Fock Transform PhL 12.31.13 1. Since the ring source includes a factor δ(ξ-ξ'), one wants the ξ variable to be "oscillatory" in the atomic form expansion. Thus, the follo wing form seems appropriate : osc expo osc chξ - cosu [Pm -1/2+iτ (chξ ) ] [exp(τu), exp(-τu) ] [ sin(m φ), cos(mφ)] The Q function is rejected because it does not vanish as r →∞ (which means ξ and u are very small, so one ends up with Qm iτ-1/2(chξ) = Qm iτ-1/2(1) = ∞. ) The function Pm -1/2+iτ (y) is called a Mehler function and is defined for y ≥1. It is of course just a Legendre function. One could also write [sh( τu), ch(τu) ] for the u atoms. In a problem which has full azimuth φ, the separation constant m is an integer. Since Pμ -ν-1(z) = Pμ ν(z) for any μ and ν, one finds that Pm -1/2-iτ(chξ) = Pm -1/2+iτ(chξ) so the sign of iτ does not matter. You can plot these Mehler functions and they are perfectly reasonable oscillatory functions. 2. There exists a transform called the generalized Mehler-Fock transform (reference in the bowl doc) g(y) = ∫0 ∞ dτ Pm -1/2+iτ(y) f(τ) // expansion f(τ ) = (τ/π) sh(πτ ) Γ(1/2-m+iτ) Γ(1/2-m-iτ ) ∫1 ∞ dy Pm -1/2+iτ(y) g(y) // projection where m is fixed. It is convenient to define s m(τ) ≡ (τ/π) sh(πτ ) Γ(1/2-m+iτ) Γ(1/2-m-iτ ) so the transform becomes g(y) = ∫0 ∞ dτ Pm -1/2+iτ(y) f(τ) // expansion f(τ ) = s m(τ) ∫1 ∞ dy Pm -1/2+iτ(y) g(y) // projection 3. Associated with any Sturm-Liouville transform is an orthogonality and completeness relation. By inserting the first in the second, and then the second in the first (of the above two equations) one finds that ∫1 ∞ dy Pm -1/2+iτ(y) Pm -1/2-iτ'(y) = δ (τ-τ') / s m(τ) // orthogonality ∫0 ∞ dτ sm(τ) Pm -1/2+iτ(y) Pm -1/2+iτ(y') = δ(y-y') // completeness 2 Replacing y = ch ξ and y' = ch ξ' we can rewrite these as ∫0 ∞ dξ shξ Pm -1/2+iτ(chξ) Pm -1/2+iτ'(chξ) = δ (τ-τ') / s m(τ) // orthogonality ∫0 ∞ dτ sm(τ) Pm -1/2+iτ(chξ) Pm -1/2+iτ(chξ') = δ(chξ-chξ') = δ (ξ-ξ')/shξ . // completeness One then has the required eigenfunction expansion to use in the Green's function source, δ(ξ-ξ') = shξ ∫0 ∞ dτ sm(τ) Pm -1/2+iτ(chξ) Pm -1/2+iτ(chξ') 4. This transform exists for any m. For m = 0 the tr ansform is called the (regular) Mehler-Fock transform. The function s m(τ) simplifies since Γ(1/2+iτ) Γ(1/2-iτ ) = π/ch(πτ) // Euler's reflection formula Γ(z) Γ(1-z) = π /sin(πz) so that s 0(τ) ≡ (τ/π) sh(πτ ) * π/ch(πτ) = τ th(πτ) . In this case we get ∫0 ∞ dξ shξ P-1/2+iτ(chξ) P-1/2+iτ'(chξ) = δ (τ-τ') /[ τ th(πτ)] orthogonality ∫0 ∞ dτ τ th(πτ) P-1/2+iτ(chξ) P-1/2+iτ(chξ') = δ(chξ-chξ') = δ (ξ-ξ')/shξ completeness so that δ(ξ-ξ') = shξ ∫0 ∞ dτ τ th(πτ) P-1/2+iτ(chξ) P-1/2+iτ(chξ') .