:py:mod:`profiley.helpers.hankel`
=================================

.. py:module:: profiley.helpers.hankel

.. autoapi-nested-parse::

   General quadrature method for Hankel transformations.
   Based on the algorithm provided in
   H. Ogata, A Numerical Integration Formula Based on the Bessel Functions,
   Publications of the Research Institute for Mathematical Sciences,
   vol. 41, no. 4, pp. 949-970, 2005.



Module Contents
---------------

Classes
~~~~~~~

.. autoapisummary::

   profiley.helpers.hankel.HankelTransform
   profiley.helpers.hankel.SphericalHankelTransform




.. py:class:: HankelTransform(nu=0, N=200, h=0.05)


   Bases: :py:obj:`object`

   The basis of the Hankel Transformation algorithm by Ogata 2005.

   This algorithm is used to solve the equation :math:`\int_0^\infty f(x) J_\nu(x) dx`
   where :math:`J_\nu(x)` is a Bessel function of the first kind of order
   :math:`nu`, and :math:`f(x)` is an arbitrary (slowly-decaying) function.

   The algorithm is presented in
   H. Ogata, A Numerical Integration Formula Based on the Bessel Functions,
   Publications of the Research Institute for Mathematical Sciences, vol. 41, no. 4, pp. 949-970, 2005.

   Parameters
   ----------
   nu : int or 0.5, optional, default = 0
       The order of the bessel function (of the first kind) J_nu(x)

   N : int, optional, default = 100
       The number of nodes in the calculation. Generally this must increase
       for a smaller value of the step-size h.

   h : float, optional, default = 0.1
       The step-size of the integration.

   .. py:method:: _psi(t)


   .. py:method:: _d_psi(t)


   .. py:method:: _weight()


   .. py:method:: _roots(N)


   .. py:method:: _j(x)


   .. py:method:: _j1(x)


   .. py:method:: _x(h)


   .. py:method:: _f(f, x)


   .. py:method:: transform(f, ret_err=True, ret_cumsum=False)

      Perform the transform of the function f

      Parameters
      ----------
      f : callable
          A function of one variable, representing :math:`f(x)`

      ret_err : boolean, optional, default = True
          Whether to return the estimated error

      ret_cumsum : boolean, optional, default = False
          Whether to return the cumulative sum



.. py:class:: SphericalHankelTransform(nu=0, *args, **kwargs)


   Bases: :py:obj:`HankelTransform`

   Perform spherical hankel transforms.

   Defined as :math:`\int_0^\infty f(x) j_\nu(x) dx`

   Note: Only does 0th-order transforms currently.

   .. py:method:: _f(f, x)


   .. py:method:: _roots(N)