Sources_For_Win/ParaView/Lib/site-packages/sympy/physics/wigner.py

# -*- coding: utf-8 -*-
r"""
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>

"""
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.function import Function
from sympy.core.numbers import (I, Integer, pi)
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import (binomial, factorial)
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.functions.special.spherical_harmonics import Ynm
from sympy.matrices.dense import zeros
from sympy.matrices.immutable import ImmutableMatrix

# This list of precomputed factorials is needed to massively
# accelerate future calculations of the various coefficients
_Factlist = [1]


def _calc_factlist(nn):
    r"""
    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]
    """
    if nn >= len(_Factlist):
        for ii in range(len(_Factlist), int(nn + 1)):
            _Factlist.append(_Factlist[ii - 1] * ii)
    return _Factlist[:int(nn) + 1]


def wigner_3j(j_1, j_2, j_3, m_1, m_2, m_3):
    r"""
    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
    """
    if int(j_1 * 2) != j_1 * 2 or int(j_2 * 2) != j_2 * 2 or \
            int(j_3 * 2) != j_3 * 2:
        raise ValueError("j values must be integer or half integer")
    if int(m_1 * 2) != m_1 * 2 or int(m_2 * 2) != m_2 * 2 or \
            int(m_3 * 2) != m_3 * 2:
        raise ValueError("m values must be integer or half integer")
    if m_1 + m_2 + m_3 != 0:
        return S.Zero
    prefid = Integer((-1) ** int(j_1 - j_2 - m_3))
    m_3 = -m_3
    a1 = j_1 + j_2 - j_3
    if a1 < 0:
        return S.Zero
    a2 = j_1 - j_2 + j_3
    if a2 < 0:
        return S.Zero
    a3 = -j_1 + j_2 + j_3
    if a3 < 0:
        return S.Zero
    if (abs(m_1) > j_1) or (abs(m_2) > j_2) or (abs(m_3) > j_3):
        return S.Zero

    maxfact = max(j_1 + j_2 + j_3 + 1, j_1 + abs(m_1), j_2 + abs(m_2),
                  j_3 + abs(m_3))
    _calc_factlist(int(maxfact))

    argsqrt = Integer(_Factlist[int(j_1 + j_2 - j_3)] *
                     _Factlist[int(j_1 - j_2 + j_3)] *
                     _Factlist[int(-j_1 + j_2 + j_3)] *
                     _Factlist[int(j_1 - m_1)] *
                     _Factlist[int(j_1 + m_1)] *
                     _Factlist[int(j_2 - m_2)] *
                     _Factlist[int(j_2 + m_2)] *
                     _Factlist[int(j_3 - m_3)] *
                     _Factlist[int(j_3 + m_3)]) / \
        _Factlist[int(j_1 + j_2 + j_3 + 1)]

    ressqrt = sqrt(argsqrt)
    if ressqrt.is_complex or ressqrt.is_infinite:
        ressqrt = ressqrt.as_real_imag()[0]

    imin = max(-j_3 + j_1 + m_2, -j_3 + j_2 - m_1, 0)
    imax = min(j_2 + m_2, j_1 - m_1, j_1 + j_2 - j_3)
    sumres = 0
    for ii in range(int(imin), int(imax) + 1):
        den = _Factlist[ii] * \
            _Factlist[int(ii + j_3 - j_1 - m_2)] * \
            _Factlist[int(j_2 + m_2 - ii)] * \
            _Factlist[int(j_1 - ii - m_1)] * \
            _Factlist[int(ii + j_3 - j_2 + m_1)] * \
            _Factlist[int(j_1 + j_2 - j_3 - ii)]
        sumres = sumres + Integer((-1) ** ii) / den

    res = ressqrt * sumres * prefid
    return res


def clebsch_gordan(j_1, j_2, j_3, m_1, m_2, m_3):
    r"""
    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
    """
    res = (-1) ** sympify(j_1 - j_2 + m_3) * sqrt(2 * j_3 + 1) * \
        wigner_3j(j_1, j_2, j_3, m_1, m_2, -m_3)
    return res


def _big_delta_coeff(aa, bb, cc, prec=None):
    r"""
    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)
    """

    if int(aa + bb - cc) != (aa + bb - cc):
        raise ValueError("j values must be integer or half integer and fulfill the triangle relation")
    if int(aa + cc - bb) != (aa + cc - bb):
        raise ValueError("j values must be integer or half integer and fulfill the triangle relation")
    if int(bb + cc - aa) != (bb + cc - aa):
        raise ValueError("j values must be integer or half integer and fulfill the triangle relation")
    if (aa + bb - cc) < 0:
        return S.Zero
    if (aa + cc - bb) < 0:
        return S.Zero
    if (bb + cc - aa) < 0:
        return S.Zero

    maxfact = max(aa + bb - cc, aa + cc - bb, bb + cc - aa, aa + bb + cc + 1)
    _calc_factlist(maxfact)

    argsqrt = Integer(_Factlist[int(aa + bb - cc)] *
                     _Factlist[int(aa + cc - bb)] *
                     _Factlist[int(bb + cc - aa)]) / \
        Integer(_Factlist[int(aa + bb + cc + 1)])

    ressqrt = sqrt(argsqrt)
    if prec:
        ressqrt = ressqrt.evalf(prec).as_real_imag()[0]
    return ressqrt


def racah(aa, bb, cc, dd, ee, ff, prec=None):
    r"""
    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
    """
    prefac = _big_delta_coeff(aa, bb, ee, prec) * \
        _big_delta_coeff(cc, dd, ee, prec) * \
        _big_delta_coeff(aa, cc, ff, prec) * \
        _big_delta_coeff(bb, dd, ff, prec)
    if prefac == 0:
        return S.Zero
    imin = max(aa + bb + ee, cc + dd + ee, aa + cc + ff, bb + dd + ff)
    imax = min(aa + bb + cc + dd, aa + dd + ee + ff, bb + cc + ee + ff)

    maxfact = max(imax + 1, aa + bb + cc + dd, aa + dd + ee + ff,
                 bb + cc + ee + ff)
    _calc_factlist(maxfact)

    sumres = 0
    for kk in range(int(imin), int(imax) + 1):
        den = _Factlist[int(kk - aa - bb - ee)] * \
            _Factlist[int(kk - cc - dd - ee)] * \
            _Factlist[int(kk - aa - cc - ff)] * \
            _Factlist[int(kk - bb - dd - ff)] * \
            _Factlist[int(aa + bb + cc + dd - kk)] * \
            _Factlist[int(aa + dd + ee + ff - kk)] * \
            _Factlist[int(bb + cc + ee + ff - kk)]
        sumres = sumres + Integer((-1) ** kk * _Factlist[kk + 1]) / den

    res = prefac * sumres * (-1) ** int(aa + bb + cc + dd)
    return res


def wigner_6j(j_1, j_2, j_3, j_4, j_5, j_6, prec=None):
    r"""
    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]_.

    """
    res = (-1) ** int(j_1 + j_2 + j_4 + j_5) * \
        racah(j_1, j_2, j_5, j_4, j_3, j_6, prec)
    return res


def wigner_9j(j_1, j_2, j_3, j_4, j_5, j_6, j_7, j_8, j_9, prec=None):
    r"""
    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]_.
    """
    imax = int(min(j_1 + j_9, j_2 + j_6, j_4 + j_8) * 2)
    imin = imax % 2
    sumres = 0
    for kk in range(imin, int(imax) + 1, 2):
        sumres = sumres + (kk + 1) * \
            racah(j_1, j_2, j_9, j_6, j_3, kk / 2, prec) * \
            racah(j_4, j_6, j_8, j_2, j_5, kk / 2, prec) * \
            racah(j_1, j_4, j_9, j_8, j_7, kk / 2, prec)
    return sumres


def gaunt(l_1, l_2, l_3, m_1, m_2, m_3, prec=None):
    r"""
    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.
    """
    if int(l_1) != l_1 or int(l_2) != l_2 or int(l_3) != l_3:
        raise ValueError("l values must be integer")
    if int(m_1) != m_1 or int(m_2) != m_2 or int(m_3) != m_3:
        raise ValueError("m values must be integer")

    sumL = l_1 + l_2 + l_3
    bigL = sumL // 2
    a1 = l_1 + l_2 - l_3
    if a1 < 0:
        return S.Zero
    a2 = l_1 - l_2 + l_3
    if a2 < 0:
        return S.Zero
    a3 = -l_1 + l_2 + l_3
    if a3 < 0:
        return S.Zero
    if sumL % 2:
        return S.Zero
    if (m_1 + m_2 + m_3) != 0:
        return S.Zero
    if (abs(m_1) > l_1) or (abs(m_2) > l_2) or (abs(m_3) > l_3):
        return S.Zero

    imin = max(-l_3 + l_1 + m_2, -l_3 + l_2 - m_1, 0)
    imax = min(l_2 + m_2, l_1 - m_1, l_1 + l_2 - l_3)

    maxfact = max(l_1 + l_2 + l_3 + 1, imax + 1)
    _calc_factlist(maxfact)

    argsqrt = (2 * l_1 + 1) * (2 * l_2 + 1) * (2 * l_3 + 1) * \
        _Factlist[l_1 - m_1] * _Factlist[l_1 + m_1] * _Factlist[l_2 - m_2] * \
        _Factlist[l_2 + m_2] * _Factlist[l_3 - m_3] * _Factlist[l_3 + m_3] / \
        (4*pi)
    ressqrt = sqrt(argsqrt)

    prefac = Integer(_Factlist[bigL] * _Factlist[l_2 - l_1 + l_3] *
                     _Factlist[l_1 - l_2 + l_3] * _Factlist[l_1 + l_2 - l_3])/ \
        _Factlist[2 * bigL + 1]/ \
        (_Factlist[bigL - l_1] *
         _Factlist[bigL - l_2] * _Factlist[bigL - l_3])

    sumres = 0
    for ii in range(int(imin), int(imax) + 1):
        den = _Factlist[ii] * _Factlist[ii + l_3 - l_1 - m_2] * \
            _Factlist[l_2 + m_2 - ii] * _Factlist[l_1 - ii - m_1] * \
            _Factlist[ii + l_3 - l_2 + m_1] * _Factlist[l_1 + l_2 - l_3 - ii]
        sumres = sumres + Integer((-1) ** ii) / den

    res = ressqrt * prefac * sumres * Integer((-1) ** (bigL + l_3 + m_1 - m_2))
    if prec is not None:
        res = res.n(prec)
    return res



class Wigner3j(Function):

    def doit(self, **hints):
        if all(obj.is_number for obj in self.args):
            return wigner_3j(*self.args)
        else:
            return self

def dot_rot_grad_Ynm(j, p, l, m, theta, phi):
    r"""
    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))

    """
    j = sympify(j)
    p = sympify(p)
    l = sympify(l)
    m = sympify(m)
    theta = sympify(theta)
    phi = sympify(phi)
    k = Dummy("k")

    def alpha(l,m,j,p,k):
        return sqrt((2*l+1)*(2*j+1)*(2*k+1)/(4*pi)) * \
                Wigner3j(j, l, k, S.Zero, S.Zero, S.Zero) * \
                Wigner3j(j, l, k, p, m, -m-p)

    return (S.NegativeOne)**(m+p) * Sum(Ynm(k, m+p, theta, phi) * alpha(l,m,j,p,k) / 2 \
        *(k**2-j**2-l**2+k-j-l), (k, abs(l-j), l+j))


def wigner_d_small(J, beta):
    """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             ⎦

    """
    M = [J-i for i in range(2*J+1)]
    d = zeros(2*J+1)
    for i, Mi in enumerate(M):
        for j, Mj in enumerate(M):

            # We get the maximum and minimum value of sigma.
            sigmamax = max([-Mi-Mj, J-Mj])
            sigmamin = min([0, J-Mi])

            dij = sqrt(factorial(J+Mi)*factorial(J-Mi) /
                       factorial(J+Mj)/factorial(J-Mj))
            terms = [(-1)**(J-Mi-s) *
                     binomial(J+Mj, J-Mi-s) *
                     binomial(J-Mj, s) *
                     cos(beta/2)**(2*s+Mi+Mj) *
                     sin(beta/2)**(2*J-2*s-Mj-Mi)
                     for s in range(sigmamin, sigmamax+1)]

            d[i, j] = dij*Add(*terms)

    return ImmutableMatrix(d)


def wigner_d(J, alpha, beta, gamma):
    """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⎠⎦

    """
    d = wigner_d_small(J, beta)
    M = [J-i for i in range(2*J+1)]
    D = [[exp(I*Mi*alpha)*d[i, j]*exp(I*Mj*gamma)
          for j, Mj in enumerate(M)] for i, Mi in enumerate(M)]
    return ImmutableMatrix(D)