Mehler Fock delta function expansion

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 following form seems appropriate : osc expo osc [Pm-1/2+iτ (chξ) ] [exp(τu), exp(-τu) ] [ sin(mφ), 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 Qmiτ-1/2(chξ) = Qmiτ-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) = Pμν(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) = !Syntax Error, Idτ Pm-1/2+iτ(y) f(τ) // expansion f(τ) = (τ/π) sh(πτ) Γ(1/2-m+iτ) Γ(1/2-m-iτ) !Syntax Error, Idy Pm-1/2+iτ(y) g(y) // projection where m is fixed. It is convenient to define sm(τ) ≡ (τ/π) sh(πτ) Γ(1/2-m+iτ) Γ(1/2-m-iτ) so the transform becomes g(y) = !Syntax Error, Idτ Pm-1/2+iτ(y) f(τ) // expansion f(τ) = sm(τ)!Syntax Error, Idy 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 !Syntax Error, Idy Pm-1/2+iτ(y) Pm-1/2-iτ'(y) = δ(τ-τ') / sm(τ) // orthogonality !Syntax Error, Idτ sm(τ) Pm-1/2+iτ(y) Pm-1/2+iτ(y') = δ(y-y') // completeness Replacing y = chξ and y' = chξ' we can rewrite these as !Syntax Error, Idξ shξ Pm-1/2+iτ(chξ) Pm-1/2+iτ'(chξ) = δ(τ-τ') / sm(τ) // orthogonality !Syntax Error, Idτ 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 !Syntax Error, Idτ sm(τ) Pm-1/2+iτ(chξ) Pm-1/2+iτ(chξ') 4. This transform exists for any m. For m = 0 the transform is called the (regular) Mehler-Fock transform. The function sm(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 s0(τ) ≡ (τ/π) sh(πτ) * π/ch(πτ) = τ th(πτ) . In this case we get !Syntax Error, Idξ shξ P-1/2+iτ(chξ) P-1/2+iτ'(chξ) = δ(τ-τ') /[ τ th(πτ)] orthogonality !Syntax Error, Idτ 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 !Syntax Error, Idτ τ th(πτ) P-1/2+iτ(chξ) P-1/2+iτ(chξ') .