.. _scf_compute_coeffs_axi: galpy.potential.scf_compute_coeffs_axi ======================================== Note: This function computes Acos and Asin as defined in `Hernquist & Ostriker (1992) `_, except that we multiply Acos by 2 such that the density from :ref:`Galpy's Hernquist Potential ` corresponds to :math:`Acos = \delta_{0n}\delta_{0l}\delta_{0m}`. Further note that this function is a specification of :ref:`scf_compute_coeffs ` where :math:`Acos_{nlm} = 0` \ at :math:`m\neq0` and :math:`Asin_{nlm} = None` For a given :math:`\rho(R, z)` we can compute :math:`Acos` and :math:`Asin` through the following equation .. math:: Acos_{nlm}= \frac{8 \pi a^3}{I_{nl}} \int_{\xi=0}^{\infty} \int_{\cos(\theta)=-1}^{1} (1 + \xi)^{2} (1 - \xi)^{-4} \rho(R, z) \Phi_{nlm}(\xi, \cos(\theta)) d\cos(\theta) d\xi \qquad Asin_{nlm}=None Where .. math:: \Phi_{nlm}(\xi, \cos(\theta)) = -\frac{\sqrt{2l + 1}}{a2^{2l + 1}} (1 + \xi)^l (1 - \xi)^{l + 1} C_{n}^{2l + 3/2}(\xi) P_{l0}(\cos(\theta)) \delta_{m0} .. 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:`m = 0` .. autofunction:: galpy.potential.scf_compute_coeffs_axi