.. _scf_compute_coeffs: galpy.potential.scf_compute_coeffs ====================================== Note: This function computes Acos and Asin as defined in Hernquist & Ostriker (1992) _, except that we multiply Acos and Asin by 2 such that the density from :ref:Galpy's Hernquist Potential  corresponds to :math:Acos = \delta_{0n}\delta_{0l}\delta_{0m} and :math:Asin = 0. For a given :math:\rho(R, z, \phi) we can compute :math:Acos and :math:Asin through the following equation .. math:: \begin{bmatrix} Acos \\ Asin \end{bmatrix}_{nlm} = \frac{4 a^3}{I_{nl}} \int_{\xi=0}^{\infty}\int_{\cos(\theta)=-1}^{1}\int_{\phi=0}^{2\pi} (1 + \xi)^{2} (1 - \xi)^{-4} \rho(R, z, \phi) \Phi_{nlm}(\xi, \cos(\theta), \phi) d\phi d\cos(\theta) d\xi Where .. math:: \Phi_{nlm}(\xi, \cos(\theta), \phi) = -\frac{\sqrt{2l + 1}}{a2^{2l + 1}} \sqrt{\frac{(l - m)!}{(l + m)!}} (1 + \xi)^l (1 - \xi)^{l + 1} C_{n}^{2l + 3/2}(\xi) P_{lm}(\cos(\theta)) \begin{bmatrix} \cos(m\phi) \\ \sin(m\phi) \end{bmatrix} .. math:: I_{nl} = - K_{nl} \frac{4\pi}{a 2^{8l + 6}} \frac{\Gamma(n + 4l + 3)}{n! (n + 2l + 3/2)[\Gamma(2l + 3/2)]^2} \qquad K_{nl} = \frac{1}{2}n(n + 4l + 3) + (l + 1)(2l + 1) :math:P_{lm} is the Associated Legendre Polynomials whereas :math:C_{n}^{\alpha} is the Gegenbauer polynomial. Also note :math:\xi = \frac{r - a}{r + a} , and :math:n, :math:l and :math:m are integers bounded by :math:0 <= n < N , :math:0 <= l < L, and :math:0 <= m <= l .. autofunction:: galpy.potential.scf_compute_coeffs