Original definition of roulette amplitudes
Using angular coordinates θ i \theta_i θ i
∂ i = ∂ ∂ θ i \partial_i=\frac{\partial}{\partial\theta_i} ∂ i = ∂ θ i ∂ we find the following expressions for the so-called Roulette amplitudes in the thin-lens and weak-field approximationClarkson (2016)
α s m = − 1 2 δ 0 s ∑ k = 0 m ( m k ) ( C s m ( k ) ∂ 1 + C s m ( k + 1 ) ∂ 2 ) ∂ 1 m − k ∂ 2 k ψ R \alpha_s^m = - \frac{1}{2^{\delta_{0s}}}
\sum_{k=0}^m\binom{m}{k}
\left(\mathcal{C}_s^{m(k)}\partial_1+\mathcal{C}_s^{m(k+1)}\partial_2\right)
\partial_1^{m-k}\partial_2^k\psi^\textrm{R} α s m = − 2 δ 0 s 1 k = 0 ∑ m ( k m ) ( C s m ( k ) ∂ 1 + C s m ( k + 1 ) ∂ 2 ) ∂ 1 m − k ∂ 2 k ψ R C s m ( k ) = 1 π ∫ − π π d ϕ sin k ϕ cos m − k + 1 ϕ cos s ϕ \mathcal{C}_s^{m(k)}=\frac{1}{\pi}\int_{-\pi}^{\pi}{\rm d}\phi\sin^k\phi\cos^{m-k+1}\phi\cos s\phi C s m ( k ) = π 1 ∫ − π π d ϕ sin k ϕ cos m − k + 1 ϕ cos s ϕ β s m = − ∑ k = 0 m ( m k ) ( S s m ( k ) ∂ 1 + S s m ( k + 1 ) ∂ 2 ) ∂ 1 m − k ∂ 2 k ψ R \beta_s^m=-\sum_{k=0}^m\binom{m}{k}\left({\mathcal{S}}_s^{m(k)}\partial_1+{\mathcal{S}}_s^{m(k+1)}\partial_2\right)\partial_1^{m-k}\partial_2^k\psi^\textrm{R} β s m = − k = 0 ∑ m ( k m ) ( S s m ( k ) ∂ 1 + S s m ( k + 1 ) ∂ 2 ) ∂ 1 m − k ∂ 2 k ψ R S s m ( k ) = 1 π ∫ − π π d ϕ sin k ϕ cos m − k + 1 ϕ sin s ϕ . \mathcal{S}_s^{m(k)}=\frac{1}{\pi}\int_{-\pi}^{\pi}{\rm d}\phi\sin^k\phi\cos^{m-k+1}\phi\sin s\phi. S s m ( k ) = π 1 ∫ − π π d ϕ sin k ϕ cos m − k + 1 ϕ sin s ϕ .
Clarkson, C. (2016). Roulettes: a weak lensing formalism for strong lensing: II. Derivation and analysis. Classical and Quantum Gravity , 33 (24), 245003. 10.1088/0264-9381/33/24/245003