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�Zddd�Zddd�Zddd�ZdS)z�Fast Hankel transforms using the FFTLog algorithm.

The implementation closely follows the Fortran code of Hamilton (2000).

added: 14/11/2020 Nicolas Tessore <n.tessore@ucl.ac.uk>
�N)�warn�)�rfft�irfft�)�loggamma�poch)�fht�ifht�	fhtoffset�c
Cs�t�|�d}|dkr$|dd}t�|�}|t�||||�}t|||||d�}t||�}	|dkrF|	t�|||||�9}	|	S)a
Compute the fast Hankel transform.

    Computes the discrete Hankel transform of a logarithmically spaced periodic
    sequence using the FFTLog algorithm [1]_, [2]_.

    Parameters
    ----------
    a : array_like (..., n)
        Real periodic input array, uniformly logarithmically spaced.  For
        multidimensional input, the transform is performed over the last axis.
    dln : float
        Uniform logarithmic spacing of the input array.
    mu : float
        Order of the Hankel transform, any positive or negative real number.
    offset : float, optional
        Offset of the uniform logarithmic spacing of the output array.
    bias : float, optional
        Exponent of power law bias, any positive or negative real number.

    Returns
    -------
    A : array_like (..., n)
        The transformed output array, which is real, periodic, uniformly
        logarithmically spaced, and of the same shape as the input array.

    See Also
    --------
    ifht : The inverse of `fht`.
    fhtoffset : Return an optimal offset for `fht`.

    Notes
    -----
    This function computes a discrete version of the Hankel transform

    .. math::

        A(k) = \int_{0}^{\infty} \! a(r) \, J_\mu(kr) \, k \, dr \;,

    where :math:`J_\mu` is the Bessel function of order :math:`\mu`.  The index
    :math:`\mu` may be any real number, positive or negative.

    The input array `a` is a periodic sequence of length :math:`n`, uniformly
    logarithmically spaced with spacing `dln`,

    .. math::

        a_j = a(r_j) \;, \quad
        r_j = r_c \exp[(j-j_c) \, \mathtt{dln}]

    centred about the point :math:`r_c`.  Note that the central index
    :math:`j_c = (n+1)/2` is half-integral if :math:`n` is even, so that
    :math:`r_c` falls between two input elements.  Similarly, the output
    array `A` is a periodic sequence of length :math:`n`, also uniformly
    logarithmically spaced with spacing `dln`

    .. math::

       A_j = A(k_j) \;, \quad
       k_j = k_c \exp[(j-j_c) \, \mathtt{dln}]

    centred about the point :math:`k_c`.

    The centre points :math:`r_c` and :math:`k_c` of the periodic intervals may
    be chosen arbitrarily, but it would be usual to choose the product
    :math:`k_c r_c = k_j r_{n-1-j} = k_{n-1-j} r_j` to be unity.  This can be
    changed using the `offset` parameter, which controls the logarithmic offset
    :math:`\log(k_c) = \mathtt{offset} - \log(r_c)` of the output array.
    Choosing an optimal value for `offset` may reduce ringing of the discrete
    Hankel transform.

    If the `bias` parameter is nonzero, this function computes a discrete
    version of the biased Hankel transform

    .. math::

        A(k) = \int_{0}^{\infty} \! a_q(r) \, (kr)^q \, J_\mu(kr) \, k \, dr

    where :math:`q` is the value of `bias`, and a power law bias
    :math:`a_q(r) = a(r) \, (kr)^{-q}` is applied to the input sequence.
    Biasing the transform can help approximate the continuous transform of
    :math:`a(r)` if there is a value :math:`q` such that :math:`a_q(r)` is
    close to a periodic sequence, in which case the resulting :math:`A(k)` will
    be close to the continuous transform.

    References
    ----------
    .. [1] Talman J. D., 1978, J. Comp. Phys., 29, 35
    .. [2] Hamilton A. J. S., 2000, MNRAS, 312, 257 (astro-ph/9905191)

    ���rrr��offset�bias��np�shape�arange�exp�fhtcoeff�_fhtq)
�a�dln�murr�n�j_c�j�u�A�r �3/usr/lib/python3/dist-packages/scipy/fft/_fftlog.pyr	s]

 r	c
Cs�t�|�d}|dkr%|dd}t�|�}|t�|||||�}t|||||d�}t||dd�}	|dkrG|	t�||||�}	|	S)a�Compute the inverse fast Hankel transform.

    Computes the discrete inverse Hankel transform of a logarithmically spaced
    periodic sequence. This is the inverse operation to `fht`.

    Parameters
    ----------
    A : array_like (..., n)
        Real periodic input array, uniformly logarithmically spaced.  For
        multidimensional input, the transform is performed over the last axis.
    dln : float
        Uniform logarithmic spacing of the input array.
    mu : float
        Order of the Hankel transform, any positive or negative real number.
    offset : float, optional
        Offset of the uniform logarithmic spacing of the output array.
    bias : float, optional
        Exponent of power law bias, any positive or negative real number.

    Returns
    -------
    a : array_like (..., n)
        The transformed output array, which is real, periodic, uniformly
        logarithmically spaced, and of the same shape as the input array.

    See Also
    --------
    fht : Definition of the fast Hankel transform.
    fhtoffset : Return an optimal offset for `ifht`.

    Notes
    -----
    This function computes a discrete version of the Hankel transform

    .. math::

        a(r) = \int_{0}^{\infty} \! A(k) \, J_\mu(kr) \, r \, dk \;,

    where :math:`J_\mu` is the Bessel function of order :math:`\mu`.  The index
    :math:`\mu` may be any real number, positive or negative.

    See `fht` for further details.

    r
rrrrT)�inverser)
rrrrrrrrrrr r r!r
�s/
r
cCsT||}}|d|d}|d|d}t�dtj|d|||dd�}	tj|ddtd�}
tj|ddtd�}|	|
jdd�<||
jdd�<t|
|d�||
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|
d�|	dt|9}	|
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|
d�d|
jd<t�
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d�s�d|t|||�|
d<|
S)z?Compute the coefficient array for a fast Hankel transform.
    rrr)�dtypeN)�outr
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(
rcCs�||}}|d|d}|d|d}tjd|}t|d|�}	t|d|�}
t|||	j|
jtj}||t�|�|S)aReturn optimal offset for a fast Hankel transform.

    Returns an offset close to `initial` that fulfils the low-ringing
    condition of [1]_ for the fast Hankel transform `fht` with logarithmic
    spacing `dln`, order `mu` and bias `bias`.

    Parameters
    ----------
    dln : float
        Uniform logarithmic spacing of the transform.
    mu : float
        Order of the Hankel transform, any positive or negative real number.
    initial : float, optional
        Initial value for the offset. Returns the closest value that fulfils
        the low-ringing condition.
    bias : float, optional
        Exponent of power law bias, any positive or negative real number.

    Returns
    -------
    offset : float
        Optimal offset of the uniform logarithmic spacing of the transform that
        fulfils a low-ringing condition.

    See also
    --------
    fht : Definition of the fast Hankel transform.

    References
    ----------
    .. [1] Hamilton A. J. S., 2000, MNRAS, 312, 257 (astro-ph/9905191)

    rry�?)rr&rr+r)�round)rr�initialrr-r.r/r0r1�zp�zm�argr r r!r�s
#rFcCs�t�|�d}t�|d�r|std�|��}d|d<n|ddkr2|r2td�|��}tj|d<t|dd�}|s?||9}n||��}t||dd�}|dddd�f}|S)zUCompute the biased fast Hankel transform.

    This is the basic FFTLog routine.
    r
rz.singular transform; consider changing the biasz6singular inverse transform; consider changing the bias)�axis.N)	rr�isinfr�copy�infr�conjr)rrr"rrr r r!r's 


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