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�8Va�z�@s�dZddlmZmZmZmZmZmZmZm	Z	m
Z
mZmZm
Z
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d�Zddd�Zddd�Zddd�ZGdd�de�Zdd�Zdd�Zdd�Z d
S)a 
Wigner, Clebsch-Gordan, Racah, and Gaunt coefficients

Collection of functions for calculating Wigner 3j, 6j, 9j,
Clebsch-Gordan, Racah as well as Gaunt coefficients exactly, all
evaluating to a rational number times the square root of a rational
number [Rasch03]_.

Please see the description of the individual functions for further
details and examples.

References
==========

.. [Regge58] 'Symmetry Properties of Clebsch-Gordan Coefficients',
  T. Regge, Nuovo Cimento, Volume 10, pp. 544 (1958)
.. [Regge59] 'Symmetry Properties of Racah Coefficients',
  T. Regge, Nuovo Cimento, Volume 11, pp. 116 (1959)
.. [Edmonds74] A. R. Edmonds. Angular momentum in quantum mechanics.
  Investigations in physics, 4.; Investigations in physics, no. 4.
  Princeton, N.J., Princeton University Press, 1957.
.. [Rasch03] J. Rasch and A. C. H. Yu, 'Efficient Storage Scheme for
  Pre-calculated Wigner 3j, 6j and Gaunt Coefficients', SIAM
  J. Sci. Comput. Volume 25, Issue 4, pp. 1416-1428 (2003)
.. [Liberatodebrito82] 'FORTRAN program for the integral of three
  spherical harmonics', A. Liberato de Brito,
  Comput. Phys. Commun., Volume 25, pp. 81-85 (1982)

Credits and Copyright
=====================

This code was taken from Sage with the permission of all authors:

https://groups.google.com/forum/#!topic/sage-devel/M4NZdu-7O38

Authors
=======

- Jens Rasch (2009-03-24): initial version for Sage

- Jens Rasch (2009-05-31): updated to sage-4.0

- Oscar Gerardo Lazo Arjona (2017-06-18): added Wigner D matrices

Copyright (C) 2008 Jens Rasch <jyr2000@gmail.com>

�)�Integer�pi�sqrt�sympify�Dummy�S�Sum�Ynm�zeros�Function�sin�cos�exp�I�	factorial�binomial�Add�ImmutableMatrix�cCsR|tt�krttt�t|d��D]
}t�t|d|�qtdt|�d�S)a1
    Function calculates a list of precomputed factorials in order to
    massively accelerate future calculations of the various
    coefficients.

    Parameters
    ==========

    nn : integer
        Highest factorial to be computed.

    Returns
    =======

    list of integers :
        The list of precomputed factorials.

    Examples
    ========

    Calculate list of factorials::

        sage: from sage.functions.wigner import _calc_factlist
        sage: _calc_factlist(10)
        [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
    rN)�len�	_Factlist�range�int�append)�nn�ii�r�6/usr/lib/python3/dist-packages/sympy/physics/wigner.py�_calc_factlist:srcCsBt|d�|dkst|d�|dkst|d�|dkr"td��t|d�|dks@t|d�|dks@t|d�|dkrDtd��|||dkrNdStdt|||��}|}|||}|dkridS|||}|dkrudS|||}	|	dkr�dSt|�|ks�t|�|ks�t|�|kr�dSt|||d|t|�|t|�|t|��}
tt|
��ttt|||�tt|||�tt|||�tt||�tt||�tt||�tt||�tt||�tt||��tt|||d�}t|�}|j�s|j	�r$|�
�d}t||||||d�}
t|||||||�}d}tt|
�t|�d�D]I}t|tt||||�tt|||�tt|||�tt||||�tt||||�}|td|�|}�qO|||}|S)a�	
    Calculate the Wigner 3j symbol `\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)`.

    Parameters
    ==========

    j_1, j_2, j_3, m_1, m_2, m_3 :
        Integer or half integer.

    Returns
    =======

    Rational number times the square root of a rational number.

    Examples
    ========

    >>> from sympy.physics.wigner import wigner_3j
    >>> wigner_3j(2, 6, 4, 0, 0, 0)
    sqrt(715)/143
    >>> wigner_3j(2, 6, 4, 0, 0, 1)
    0

    It is an error to have arguments that are not integer or half
    integer values::

        sage: wigner_3j(2.1, 6, 4, 0, 0, 0)
        Traceback (most recent call last):
        ...
        ValueError: j values must be integer or half integer
        sage: wigner_3j(2, 6, 4, 1, 0, -1.1)
        Traceback (most recent call last):
        ...
        ValueError: m values must be integer or half integer

    Notes
    =====

    The Wigner 3j symbol obeys the following symmetry rules:

    - invariant under any permutation of the columns (with the
      exception of a sign change where `J:=j_1+j_2+j_3`):

      .. math::

         \begin{aligned}
         \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)
          &=\operatorname{Wigner3j}(j_3,j_1,j_2,m_3,m_1,m_2) \\
          &=\operatorname{Wigner3j}(j_2,j_3,j_1,m_2,m_3,m_1) \\
          &=(-1)^J \operatorname{Wigner3j}(j_3,j_2,j_1,m_3,m_2,m_1) \\
          &=(-1)^J \operatorname{Wigner3j}(j_1,j_3,j_2,m_1,m_3,m_2) \\
          &=(-1)^J \operatorname{Wigner3j}(j_2,j_1,j_3,m_2,m_1,m_3)
         \end{aligned}

    - invariant under space inflection, i.e.

      .. math::

         \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)
         =(-1)^J \operatorname{Wigner3j}(j_1,j_2,j_3,-m_1,-m_2,-m_3)

    - symmetric with respect to the 72 additional symmetries based on
      the work by [Regge58]_

    - zero for `j_1`, `j_2`, `j_3` not fulfilling triangle relation

    - zero for `m_1 + m_2 + m_3 \neq 0`

    - zero for violating any one of the conditions
      `j_1 \ge |m_1|`,  `j_2 \ge |m_2|`,  `j_3 \ge |m_3|`

    Algorithm
    =========

    This function uses the algorithm of [Edmonds74]_ to calculate the
    value of the 3j symbol exactly. Note that the formula contains
    alternating sums over large factorials and is therefore unsuitable
    for finite precision arithmetic and only useful for a computer
    algebra system [Rasch03]_.

    Authors
    =======

    - Jens Rasch (2009-03-24): initial version
    �z(j values must be integer or half integerz(m values must be integer or half integerr���r)
r�
ValueErrorr�abs�maxrrr�
is_complex�is_infinite�as_real_imag�minr)�j_1�j_2�j_3�m_1�m_2�m_3�prefid�a1�a2�a3�maxfact�argsqrt�ressqrt�imin�imax�sumresr�den�resrrr�	wigner_3j[s�(V($$
���������	� �����r:cCs<dt|||�td|d�t||||||�}|S)a�
    Calculates the Clebsch-Gordan coefficient.
    `\left\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \right\rangle`.

    The reference for this function is [Edmonds74]_.

    Parameters
    ==========

    j_1, j_2, j_3, m_1, m_2, m_3 :
        Integer or half integer.

    Returns
    =======

    Rational number times the square root of a rational number.

    Examples
    ========

    >>> from sympy import S
    >>> from sympy.physics.wigner import clebsch_gordan
    >>> clebsch_gordan(S(3)/2, S(1)/2, 2, S(3)/2, S(1)/2, 2)
    1
    >>> clebsch_gordan(S(3)/2, S(1)/2, 1, S(3)/2, -S(1)/2, 1)
    sqrt(3)/2
    >>> clebsch_gordan(S(3)/2, S(1)/2, 1, -S(1)/2, S(1)/2, 0)
    -sqrt(2)/2

    Notes
    =====

    The Clebsch-Gordan coefficient will be evaluated via its relation
    to Wigner 3j symbols:

    .. math::

        \left\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \right\rangle
        =(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1}
        \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,-m_3)

    See also the documentation on Wigner 3j symbols which exhibit much
    higher symmetry relations than the Clebsch-Gordan coefficient.

    Authors
    =======

    - Jens Rasch (2009-03-24): initial version
    r rr)rrr:)r(r)r*r+r,r-r9rrr�clebsch_gordan�s"2�r;NcCs`t|||�|||krtd��t|||�|||kr$td��t|||�|||kr6td��|||dkr@dS|||dkrJdS|||dkrTdSt||||||||||||d�}t|�ttt|||�tt|||�tt|||��ttt|||d��}t|�}|r�|�|���d}|S)a�
    Calculates the Delta coefficient of the 3 angular momenta for
    Racah symbols. Also checks that the differences are of integer
    value.

    Parameters
    ==========

    aa :
        First angular momentum, integer or half integer.
    bb :
        Second angular momentum, integer or half integer.
    cc :
        Third angular momentum, integer or half integer.
    prec :
        Precision of the ``sqrt()`` calculation.

    Returns
    =======

    double : Value of the Delta coefficient.

    Examples
    ========

        sage: from sage.functions.wigner import _big_delta_coeff
        sage: _big_delta_coeff(1,1,1)
        1/2*sqrt(1/6)
    zJj values must be integer or half integer and fulfill the triangle relationrr)	rr!r#rrrr�evalfr&)�aa�bb�cc�precr2r3r4rrr�_big_delta_coeff!s22���rAcCs�t||||�t||||�t||||�t||||�}|dkr"dSt||||||||||||�}t||||||||||||�}	t|	d||||||||||||�}
t|
�d}tt|�t|	�d�D]l}tt||||�tt||||�tt||||�tt||||�tt|||||�tt|||||�tt|||||�}
|td|t|d�|
}q}||dt||||�}|S)a�
    Calculate the Racah symbol `W(a,b,c,d;e,f)`.

    Parameters
    ==========

    a, ..., f :
        Integer or half integer.
    prec :
        Precision, default: ``None``. Providing a precision can
        drastically speed up the calculation.

    Returns
    =======

    Rational number times the square root of a rational number
    (if ``prec=None``), or real number if a precision is given.

    Examples
    ========

    >>> from sympy.physics.wigner import racah
    >>> racah(3,3,3,3,3,3)
    -1/14

    Notes
    =====

    The Racah symbol is related to the Wigner 6j symbol:

    .. math::

       \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)
       =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)

    Please see the 6j symbol for its much richer symmetries and for
    additional properties.

    Algorithm
    =========

    This function uses the algorithm of [Edmonds74]_ to calculate the
    value of the 6j symbol exactly. Note that the formula contains
    alternating sums over large factorials and is therefore unsuitable
    for finite precision arithmetic and only useful for a computer
    algebra system [Rasch03]_.

    Authors
    =======

    - Jens Rasch (2009-03-24): initial version
    rrr )rAr#r'rrrrr)r=r>r?�dd�ee�ffr@�prefacr5r6r2r7�kkr8r9rrr�racah[sB5���.0$�������" rGc	Cs0dt||||�t|||||||�}|S)a+
    Calculate the Wigner 6j symbol `\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)`.

    Parameters
    ==========

    j_1, ..., j_6 :
        Integer or half integer.
    prec :
        Precision, default: ``None``. Providing a precision can
        drastically speed up the calculation.

    Returns
    =======

    Rational number times the square root of a rational number
    (if ``prec=None``), or real number if a precision is given.

    Examples
    ========

    >>> from sympy.physics.wigner import wigner_6j
    >>> wigner_6j(3,3,3,3,3,3)
    -1/14
    >>> wigner_6j(5,5,5,5,5,5)
    1/52

    It is an error to have arguments that are not integer or half
    integer values or do not fulfill the triangle relation::

        sage: wigner_6j(2.5,2.5,2.5,2.5,2.5,2.5)
        Traceback (most recent call last):
        ...
        ValueError: j values must be integer or half integer and fulfill the triangle relation
        sage: wigner_6j(0.5,0.5,1.1,0.5,0.5,1.1)
        Traceback (most recent call last):
        ...
        ValueError: j values must be integer or half integer and fulfill the triangle relation

    Notes
    =====

    The Wigner 6j symbol is related to the Racah symbol but exhibits
    more symmetries as detailed below.

    .. math::

       \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)
        =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)

    The Wigner 6j symbol obeys the following symmetry rules:

    - Wigner 6j symbols are left invariant under any permutation of
      the columns:

      .. math::

         \begin{aligned}
         \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)
          &=\operatorname{Wigner6j}(j_3,j_1,j_2,j_6,j_4,j_5) \\
          &=\operatorname{Wigner6j}(j_2,j_3,j_1,j_5,j_6,j_4) \\
          &=\operatorname{Wigner6j}(j_3,j_2,j_1,j_6,j_5,j_4) \\
          &=\operatorname{Wigner6j}(j_1,j_3,j_2,j_4,j_6,j_5) \\
          &=\operatorname{Wigner6j}(j_2,j_1,j_3,j_5,j_4,j_6)
         \end{aligned}

    - They are invariant under the exchange of the upper and lower
      arguments in each of any two columns, i.e.

      .. math::

         \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)
          =\operatorname{Wigner6j}(j_1,j_5,j_6,j_4,j_2,j_3)
          =\operatorname{Wigner6j}(j_4,j_2,j_6,j_1,j_5,j_3)
          =\operatorname{Wigner6j}(j_4,j_5,j_3,j_1,j_2,j_6)

    - additional 6 symmetries [Regge59]_ giving rise to 144 symmetries
      in total

    - only non-zero if any triple of `j`'s fulfill a triangle relation

    Algorithm
    =========

    This function uses the algorithm of [Edmonds74]_ to calculate the
    value of the 6j symbol exactly. Note that the formula contains
    alternating sums over large factorials and is therefore unsuitable
    for finite precision arithmetic and only useful for a computer
    algebra system [Rasch03]_.

    r )rrG)r(r)r*�j_4�j_5�j_6r@r9rrr�	wigner_6j�s\�rKc
Cs�tt||||||�d�}
|
d}d}t|t|
�dd�D],}
||
dt||||||
d|	�t||||||
d|	�t||||||
d|	�}q |S)a?
    Calculate the Wigner 9j symbol
    `\operatorname{Wigner9j}(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9)`.

    Parameters
    ==========

    j_1, ..., j_9 :
        Integer or half integer.
    prec : precision, default
        ``None``. Providing a precision can
        drastically speed up the calculation.

    Returns
    =======

    Rational number times the square root of a rational number
    (if ``prec=None``), or real number if a precision is given.

    Examples
    ========

    >>> from sympy.physics.wigner import wigner_9j
    >>> wigner_9j(1,1,1, 1,1,1, 1,1,0 ,prec=64) # ==1/18
    0.05555555...

    >>> wigner_9j(1/2,1/2,0, 1/2,3/2,1, 0,1,1 ,prec=64) # ==1/6
    0.1666666...

    It is an error to have arguments that are not integer or half
    integer values or do not fulfill the triangle relation::

        sage: wigner_9j(0.5,0.5,0.5, 0.5,0.5,0.5, 0.5,0.5,0.5,prec=64)
        Traceback (most recent call last):
        ...
        ValueError: j values must be integer or half integer and fulfill the triangle relation
        sage: wigner_9j(1,1,1, 0.5,1,1.5, 0.5,1,2.5,prec=64)
        Traceback (most recent call last):
        ...
        ValueError: j values must be integer or half integer and fulfill the triangle relation

    Algorithm
    =========

    This function uses the algorithm of [Edmonds74]_ to calculate the
    value of the 3j symbol exactly. Note that the formula contains
    alternating sums over large factorials and is therefore unsuitable
    for finite precision arithmetic and only useful for a computer
    algebra system [Rasch03]_.
    rrr)rr'rrG)r(r)r*rHrIrJ�j_7�j_8�j_9r@r6r5r7rFrrr�	wigner_9j
s 3���rOcCst|�|kst|�|kst|�|krtd��t|�|ks(t|�|ks(t|�|kr,td��|||}|d}|||}	|	dkrBdS|||}
|
dkrNdS|||}|dkr[dS|dradS|||dkrkdSt|�|ks}t|�|ks}t|�|krdSt||||||d�}t|||||||�}
t|||d|
d�}t|�d|dd|dd|dt||t||t||t||t||t||dt}t|�}t	t|t|||t|||t|||�td|dt||t||t||}d}t
t|�t|
�d�D]?}t|t||||t|||t|||t||||t||||}|t	d|�|}�q1|||t	d||||�}|du�r�|�|�}|S)	a*
    Calculate the Gaunt coefficient.

    Explanation
    ===========

    The Gaunt coefficient is defined as the integral over three
    spherical harmonics:

    .. math::

        \begin{aligned}
        \operatorname{Gaunt}(l_1,l_2,l_3,m_1,m_2,m_3)
        &=\int Y_{l_1,m_1}(\Omega)
         Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) \,d\Omega \\
        &=\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}}
         \operatorname{Wigner3j}(l_1,l_2,l_3,0,0,0)
         \operatorname{Wigner3j}(l_1,l_2,l_3,m_1,m_2,m_3)
        \end{aligned}

    Parameters
    ==========

    l_1, l_2, l_3, m_1, m_2, m_3 :
        Integer.
    prec - precision, default: ``None``.
        Providing a precision can
        drastically speed up the calculation.

    Returns
    =======

    Rational number times the square root of a rational number
    (if ``prec=None``), or real number if a precision is given.

    Examples
    ========

    >>> from sympy.physics.wigner import gaunt
    >>> gaunt(1,0,1,1,0,-1)
    -1/(2*sqrt(pi))
    >>> gaunt(1000,1000,1200,9,3,-12).n(64)
    0.00689500421922113448...

    It is an error to use non-integer values for `l` and `m`::

        sage: gaunt(1.2,0,1.2,0,0,0)
        Traceback (most recent call last):
        ...
        ValueError: l values must be integer
        sage: gaunt(1,0,1,1.1,0,-1.1)
        Traceback (most recent call last):
        ...
        ValueError: m values must be integer

    Notes
    =====

    The Gaunt coefficient obeys the following symmetry rules:

    - invariant under any permutation of the columns

      .. math::
        \begin{aligned}
          Y(l_1,l_2,l_3,m_1,m_2,m_3)
          &=Y(l_3,l_1,l_2,m_3,m_1,m_2) \\
          &=Y(l_2,l_3,l_1,m_2,m_3,m_1) \\
          &=Y(l_3,l_2,l_1,m_3,m_2,m_1) \\
          &=Y(l_1,l_3,l_2,m_1,m_3,m_2) \\
          &=Y(l_2,l_1,l_3,m_2,m_1,m_3)
        \end{aligned}

    - invariant under space inflection, i.e.

      .. math::
          Y(l_1,l_2,l_3,m_1,m_2,m_3)
          =Y(l_1,l_2,l_3,-m_1,-m_2,-m_3)

    - symmetric with respect to the 72 Regge symmetries as inherited
      for the `3j` symbols [Regge58]_

    - zero for `l_1`, `l_2`, `l_3` not fulfilling triangle relation

    - zero for violating any one of the conditions: `l_1 \ge |m_1|`,
      `l_2 \ge |m_2|`, `l_3 \ge |m_3|`

    - non-zero only for an even sum of the `l_i`, i.e.
      `L = l_1 + l_2 + l_3 = 2n` for `n` in `\mathbb{N}`

    Algorithms
    ==========

    This function uses the algorithm of [Liberatodebrito82]_ to
    calculate the value of the Gaunt coefficient exactly. Note that
    the formula contains alternating sums over large factorials and is
    therefore unsuitable for finite precision arithmetic and only
    useful for a computer algebra system [Rasch03]_.

    Authors
    =======

    Jens Rasch (2009-03-24): initial version for Sage.
    zl values must be integerzm values must be integerrrr�r N)rr!r"r#r'rrrrrr�n)�l_1�l_2�l_3r+r,r-r@�sumL�bigLr/r0r1r5r6r2r3r4rEr7rr8r9rrr�gauntKs�$h$$ "
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rWc@seZdZdd�ZdS)�Wigner3jcKs"tdd�|jD��rt|j�S|S)Ncss�|]}|jVqdS�N)�	is_number)�.0�objrrr�	<genexpr>�s�z Wigner3j.doit.<locals>.<genexpr>)�all�argsr:)�self�hintsrrr�doit�s
z
Wigner3j.doitN)�__name__�
__module__�__qualname__rbrrrrrX�srXc	Cs�t|�}t|�}t|�}t|�}t|�}t|�}td�}dd�}tj||tt|||||�||||||�d|d|d|d||||t||�||f�S)a)
    Returns dot product of rotational gradients of spherical harmonics.

    Explanation
    ===========

    This function returns the right hand side of the following expression:

    .. math ::
        \vec{R}Y{_j^{p}} \cdot \vec{R}Y{_l^{m}} = (-1)^{m+p}
        \sum\limits_{k=|l-j|}^{l+j}Y{_k^{m+p}}  * \alpha_{l,m,j,p,k} *
        \frac{1}{2} (k^2-j^2-l^2+k-j-l)


    Arguments
    =========

    j, p, l, m .... indices in spherical harmonics (expressions or integers)
    theta, phi .... angle arguments in spherical harmonics

    Example
    =======

    >>> from sympy import symbols
    >>> from sympy.physics.wigner import dot_rot_grad_Ynm
    >>> theta, phi = symbols("theta phi")
    >>> dot_rot_grad_Ynm(3, 2, 2, 0, theta, phi).doit()
    3*sqrt(55)*Ynm(5, 2, theta, phi)/(11*sqrt(pi))

    �kc	Ss`td|dd|dd|ddt�t|||tjtjtj�t|||||||�S)NrrrP)rrrXr�Zero)�l�m�j�prfrrr�alphas
.��zdot_rot_grad_Ynm.<locals>.alphar)rrr�NegativeOnerr	r")rjrkrhri�theta�phirfrlrrr�dot_rot_grad_Ynm�s2"��rpc
s��fdd�td�d�D�}td�d�}t|�D]U\}�t|�D]L\}�t����g�}td��g�}tt���t���t���t����}����fdd�t||d�D�}	|t|	�|||f<q#qt|�S)u�Return the small Wigner d matrix for angular momentum J.

    Explanation
    ===========

    J : An integer, half-integer, or sympy symbol for the total angular
        momentum of the angular momentum space being rotated.
    beta : A real number representing the Euler angle of rotation about
        the so-called line of nodes. See [Edmonds74]_.

    Returns
    =======

    A matrix representing the corresponding Euler angle rotation( in the basis
    of eigenvectors of `J_z`).

    .. math ::
        \mathcal{d}_{\beta} = \exp\big( \frac{i\beta}{\hbar} J_y\big)

    The components are calculated using the general form [Edmonds74]_,
    equation 4.1.15.

    Examples
    ========

    >>> from sympy import Integer, symbols, pi, pprint
    >>> from sympy.physics.wigner import wigner_d_small
    >>> half = 1/Integer(2)
    >>> beta = symbols("beta", real=True)
    >>> pprint(wigner_d_small(half, beta), use_unicode=True)
    ⎡   ⎛β⎞      ⎛β⎞⎤
    ⎢cos⎜─⎟   sin⎜─⎟⎥
    ⎢   ⎝2⎠      ⎝2⎠⎥
    ⎢               ⎥
    ⎢    ⎛β⎞     ⎛β⎞⎥
    ⎢-sin⎜─⎟  cos⎜─⎟⎥
    ⎣    ⎝2⎠     ⎝2⎠⎦

    >>> pprint(wigner_d_small(2*half, beta), use_unicode=True)
    ⎡        2⎛β⎞              ⎛β⎞    ⎛β⎞           2⎛β⎞     ⎤
    ⎢     cos ⎜─⎟        √2⋅sin⎜─⎟⋅cos⎜─⎟        sin ⎜─⎟     ⎥
    ⎢         ⎝2⎠              ⎝2⎠    ⎝2⎠            ⎝2⎠     ⎥
    ⎢                                                        ⎥
    ⎢       ⎛β⎞    ⎛β⎞       2⎛β⎞      2⎛β⎞        ⎛β⎞    ⎛β⎞⎥
    ⎢-√2⋅sin⎜─⎟⋅cos⎜─⎟  - sin ⎜─⎟ + cos ⎜─⎟  √2⋅sin⎜─⎟⋅cos⎜─⎟⎥
    ⎢       ⎝2⎠    ⎝2⎠        ⎝2⎠       ⎝2⎠        ⎝2⎠    ⎝2⎠⎥
    ⎢                                                        ⎥
    ⎢        2⎛β⎞               ⎛β⎞    ⎛β⎞          2⎛β⎞     ⎥
    ⎢     sin ⎜─⎟        -√2⋅sin⎜─⎟⋅cos⎜─⎟       cos ⎜─⎟     ⎥
    ⎣         ⎝2⎠               ⎝2⎠    ⎝2⎠           ⎝2⎠     ⎦

    From table 4 in [Edmonds74]_

    >>> pprint(wigner_d_small(half, beta).subs({beta:pi/2}), use_unicode=True)
    ⎡ √2   √2⎤
    ⎢ ──   ──⎥
    ⎢ 2    2 ⎥
    ⎢        ⎥
    ⎢-√2   √2⎥
    ⎢────  ──⎥
    ⎣ 2    2 ⎦

    >>> pprint(wigner_d_small(2*half, beta).subs({beta:pi/2}),
    ... use_unicode=True)
    ⎡       √2      ⎤
    ⎢1/2    ──   1/2⎥
    ⎢       2       ⎥
    ⎢               ⎥
    ⎢-√2         √2 ⎥
    ⎢────   0    ── ⎥
    ⎢ 2          2  ⎥
    ⎢               ⎥
    ⎢      -√2      ⎥
    ⎢1/2   ────  1/2⎥
    ⎣       2       ⎦

    >>> pprint(wigner_d_small(3*half, beta).subs({beta:pi/2}),
    ... use_unicode=True)
    ⎡ √2    √6    √6   √2⎤
    ⎢ ──    ──    ──   ──⎥
    ⎢ 4     4     4    4 ⎥
    ⎢                    ⎥
    ⎢-√6   -√2    √2   √6⎥
    ⎢────  ────   ──   ──⎥
    ⎢ 4     4     4    4 ⎥
    ⎢                    ⎥
    ⎢ √6   -√2   -√2   √6⎥
    ⎢ ──   ────  ────  ──⎥
    ⎢ 4     4     4    4 ⎥
    ⎢                    ⎥
    ⎢-√2    √6   -√6   √2⎥
    ⎢────   ──   ────  ──⎥
    ⎣ 4     4     4    4 ⎦

    >>> pprint(wigner_d_small(4*half, beta).subs({beta:pi/2}),
    ... use_unicode=True)
    ⎡             √6            ⎤
    ⎢1/4   1/2    ──   1/2   1/4⎥
    ⎢             4             ⎥
    ⎢                           ⎥
    ⎢-1/2  -1/2   0    1/2   1/2⎥
    ⎢                           ⎥
    ⎢ √6                     √6 ⎥
    ⎢ ──    0    -1/2   0    ── ⎥
    ⎢ 4                      4  ⎥
    ⎢                           ⎥
    ⎢-1/2  1/2    0    -1/2  1/2⎥
    ⎢                           ⎥
    ⎢             √6            ⎥
    ⎢1/4   -1/2   ──   -1/2  1/4⎥
    ⎣             4             ⎦

    c�g|]}�|�qSrr�r[�i��Jrr�
<listcomp>��z"wigner_d_small.<locals>.<listcomp>rrrcs�g|]<}d��|t����|�t��|�t�d�d|��t�d�d�d|���qS)r r)rr
r)r[�s�ru�Mi�Mj�betarrrv�s����"�)	rr
�	enumerater#r'rrrr)
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�
���r�csLt�|���fdd�td�d�D������fdd�t��D�}t|�S)u�Return the Wigner D matrix for angular momentum J.

    Explanation
    ===========

    J :
        An integer, half-integer, or sympy symbol for the total angular
        momentum of the angular momentum space being rotated.
    alpha, beta, gamma - Real numbers representing the Euler.
        Angles of rotation about the so-called vertical, line of nodes, and
        figure axes. See [Edmonds74]_.

    Returns
    =======

    A matrix representing the corresponding Euler angle rotation( in the basis
    of eigenvectors of `J_z`).

    .. math ::
        \mathcal{D}_{\alpha \beta \gamma} =
        \exp\big( \frac{i\alpha}{\hbar} J_z\big)
        \exp\big( \frac{i\beta}{\hbar} J_y\big)
        \exp\big( \frac{i\gamma}{\hbar} J_z\big)

    The components are calculated using the general form [Edmonds74]_,
    equation 4.1.12.

    Examples
    ========

    The simplest possible example:

    >>> from sympy.physics.wigner import wigner_d
    >>> from sympy import Integer, symbols, pprint
    >>> half = 1/Integer(2)
    >>> alpha, beta, gamma = symbols("alpha, beta, gamma", real=True)
    >>> pprint(wigner_d(half, alpha, beta, gamma), use_unicode=True)
    ⎡  ⅈ⋅α  ⅈ⋅γ             ⅈ⋅α  -ⅈ⋅γ         ⎤
    ⎢  ───  ───             ───  ─────        ⎥
    ⎢   2    2     ⎛β⎞       2     2      ⎛β⎞ ⎥
    ⎢ ℯ   ⋅ℯ   ⋅cos⎜─⎟     ℯ   ⋅ℯ     ⋅sin⎜─⎟ ⎥
    ⎢              ⎝2⎠                    ⎝2⎠ ⎥
    ⎢                                         ⎥
    ⎢  -ⅈ⋅α   ⅈ⋅γ          -ⅈ⋅α   -ⅈ⋅γ        ⎥
    ⎢  ─────  ───          ─────  ─────       ⎥
    ⎢    2     2     ⎛β⎞     2      2      ⎛β⎞⎥
    ⎢-ℯ     ⋅ℯ   ⋅sin⎜─⎟  ℯ     ⋅ℯ     ⋅cos⎜─⎟⎥
    ⎣                ⎝2⎠                   ⎝2⎠⎦

    crqrrrrrtrrrv�rwzwigner_d.<locals>.<listcomp>rrcs.g|]\�������fdd�t��D��qS)cs<g|]\}}tt�����|ftt|���qSr)rr)r[rjr{)rzrlr�gammarsrrrv�s0�z'wigner_d.<locals>.<listcomp>.<listcomp>)r})r[)r~rlrr�)rzrsrrv�s
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