calc_sn
=======

.. py:function:: luminet.black_hole_math.calc_sn(p: float, angle: float, bh_mass: float, incl: float, order: int = 0) -> float

   Calculate the elliptic function :math:`\text{sn}`

   For direct images, this is:

   .. math::

       \text{sn} \left( \frac{\gamma}{2 \sqrt{P/Q}} + F(\zeta_{\infty}, k) \right)

   For higher order images, this is:

   .. math::

       \text{sn} \left( \frac{\gamma - 2n\pi}{2 \sqrt{P/Q}} - F(\zeta_{\infty}, k) + 2K(k) \right)

   Here, :math:`F` is the incomplete elliptic integral of the first kind,
   and :math:`K` is the complete elliptic integral of the first kind.
   Elliptic integrals and elliptic functions are related:

   .. math::

      u &= F(\phi,m) \\
      \text{sn}(u|m) &= sin(\phi)


   .. attention::

      Note that ``scipy`` uses the modulus :math:`m = k^2` in the elliptic integrals,
      not the modulus :math:`k`.

   :Parameters: * **p** (*float*) -- periastron distance
                * **angle** (*float*) -- Angle in the black hole frame :math:`\alpha`
                * **bh_mass** (*float*) -- Black hole mass
                * **incl** (*float*) -- Inclination of the observer :math:`\theta_0`
                * **order** (*int*) -- Order of the image. Default is :math:`0` (direct image).

   :returns: *float* -- Value of the elliptic integral :math:`\text{sn}`