
    sg*                         d dl mZ d dlmZmZ d dlmZmZ d dlm	Z	 d dl
mZ d dlmZ d dlmZ d dlmZmZ d d	lmZ d d
lmZ d dlmZmZmZ  ed      Z G d de      Zd Z G d de      Zy)    )Expr)FunctionArgumentIndexError)Ipi)S)Dummy)assoc_legendre)	factorial)Abs	conjugate)exp)sqrt)sincoscotxc                   T    e Zd ZdZed        Zd ZddZd Zd Z	d Z
d Zdd	Zd
 Zy)Ynma4  
    Spherical harmonics defined as

    .. math::
        Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}
                                  \exp(i m \varphi)
                                  \mathrm{P}_n^m\left(\cos(\theta)\right)

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

    ``Ynm()`` gives the spherical harmonic function of order $n$ and $m$
    in $\theta$ and $\varphi$, $Y_n^m(\theta, \varphi)$. The four
    parameters are as follows: $n \geq 0$ an integer and $m$ an integer
    such that $-n \leq m \leq n$ holds. The two angles are real-valued
    with $\theta \in [0, \pi]$ and $\varphi \in [0, 2\pi]$.

    Examples
    ========

    >>> from sympy import Ynm, Symbol, simplify
    >>> from sympy.abc import n,m
    >>> theta = Symbol("theta")
    >>> phi = Symbol("phi")

    >>> Ynm(n, m, theta, phi)
    Ynm(n, m, theta, phi)

    Several symmetries are known, for the order:

    >>> Ynm(n, -m, theta, phi)
    (-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

    As well as for the angles:

    >>> Ynm(n, m, -theta, phi)
    Ynm(n, m, theta, phi)

    >>> Ynm(n, m, theta, -phi)
    exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

    For specific integers $n$ and $m$ we can evaluate the harmonics
    to more useful expressions:

    >>> simplify(Ynm(0, 0, theta, phi).expand(func=True))
    1/(2*sqrt(pi))

    >>> simplify(Ynm(1, -1, theta, phi).expand(func=True))
    sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))

    >>> simplify(Ynm(1, 0, theta, phi).expand(func=True))
    sqrt(3)*cos(theta)/(2*sqrt(pi))

    >>> simplify(Ynm(1, 1, theta, phi).expand(func=True))
    -sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))

    >>> simplify(Ynm(2, -2, theta, phi).expand(func=True))
    sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))

    >>> simplify(Ynm(2, -1, theta, phi).expand(func=True))
    sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))

    >>> simplify(Ynm(2, 0, theta, phi).expand(func=True))
    sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))

    >>> simplify(Ynm(2, 1, theta, phi).expand(func=True))
    -sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))

    >>> simplify(Ynm(2, 2, theta, phi).expand(func=True))
    sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))

    We can differentiate the functions with respect
    to both angles:

    >>> from sympy import Ynm, Symbol, diff
    >>> from sympy.abc import n,m
    >>> theta = Symbol("theta")
    >>> phi = Symbol("phi")

    >>> diff(Ynm(n, m, theta, phi), theta)
    m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)

    >>> diff(Ynm(n, m, theta, phi), phi)
    I*m*Ynm(n, m, theta, phi)

    Further we can compute the complex conjugation:

    >>> from sympy import Ynm, Symbol, conjugate
    >>> from sympy.abc import n,m
    >>> theta = Symbol("theta")
    >>> phi = Symbol("phi")

    >>> conjugate(Ynm(n, m, theta, phi))
    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

    To get back the well known expressions in spherical
    coordinates, we use full expansion:

    >>> from sympy import Ynm, Symbol, expand_func
    >>> from sympy.abc import n,m
    >>> theta = Symbol("theta")
    >>> phi = Symbol("phi")

    >>> expand_func(Ynm(n, m, theta, phi))
    sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))

    See Also
    ========

    Ynm_c, Znm

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
    .. [4] https://dlmf.nist.gov/14.30

    c                 V   |j                         r>| }t        j                  |z  t        dt        z  |z  |z        z  t        ||||      z  S |j                         r| }t        ||||      S |j                         r*| }t        dt        z  |z  |z        t        ||||      z  S y )N)could_extract_minus_signr   NegativeOner   r   r   )clsnmthetaphis        ^/var/www/html/venv/lib/python3.12/site-packages/sympy/functions/special/spherical_harmonics.pyevalzYnm.eval   s     %%'A==!#c"Q$q&*o5Aq%8MMM))+FEq!UC(('')$Cr!tAvcz?SAuc%::: *    c                 X   | j                   \  }}}}t        d|z  dz   dt        z  z  t        ||z
        z  t        ||z         z        t	        t
        |z  |z        z  t        ||t        |            z  }|j                  t        t        |      dz   dz         t        |            S N         )
argsr   r   r   r   r   r
   r   subsr   )selfhintsr   r   r   r   rvs          r   _eval_expand_funczYnm._eval_expand_func   s    991eSAaC!Gad#iA&66yQ7GGHAaCG-aCJ?@ wwtSZ]NQ./U<<r!   c                    |dk(  rt        | |      |dk(  rt        | |      |dk(  rl| j                  \  }}}}|t        |      z  t        ||||      z  t	        ||z
  ||z   dz   z        t        t         |z        z  t        ||dz   ||      z  z   S |dk(  r)| j                  \  }}}}t        |z  t        ||||      z  S t        | |      )Nr%   r$      r&   )r   r'   r   r   r   r   r   )r)   argindexr   r   r   r   s         r   fdiffz	Ynm.fdiff   s    q=$T844]$T844]#yyAq%E
NSAuc%::!a%!a%!),-QBsF;c!QUESV>WWX Y]#yyAq%q53q!UC000$T844r!   c                 &    | j                  d      S )NTfunc)expandr)   r   r   r   r   kwargss         r   _eval_rewrite_as_polynomialzYnm._eval_rewrite_as_polynomial   s     {{{%%r!   c                 ,    | j                  t              S N)rewriter   r5   s         r   _eval_rewrite_as_sinzYnm._eval_rewrite_as_sin   s    ||C  r!   c                     ddl m}m}  || j                  d            }|j	                  t        t        |            t        |      i      } | ||            S )Nr   )simplifytrigsimpTr2   )sympy.simplifyr=   r>   r4   xreplacer   r   )	r)   r   r   r   r   r6   r=   r>   terms	            r   _eval_rewrite_as_coszYnm._eval_rewrite_as_cos   sJ    5 ./}}c#e*oc%j9:''r!   c                 v    | j                   \  }}}}t        j                  |z  | j                  || ||      z  S r9   )r'   r   r   r3   )r)   r   r   r   r   s        r   _eval_conjugatezYnm._eval_conjugate   s9    991eS}}a$))Ar5#">>>r!   c                    | j                   \  }}}}t        d|z  dz   dt        z  z  t        ||z
        z  t        ||z         z        t	        ||z        z  t        ||t	        |            z  }t        d|z  dz   dt        z  z  t        ||z
        z  t        ||z         z        t        ||z        z  t        ||t	        |            z  }||fS r#   )r'   r   r   r   r   r
   r   )	r)   deepr*   r   r   r   r   reims	            r   as_real_imagzYnm.as_real_imag   s    991eSAaC!Gad#iA&66yQ7GGH!C%j)!QE
;<AaC!Gad#iA&66yQ7GGH!C%j)!QE
;<Bxr!   c                    ddl m}m} | j                  d   j	                  |      }| j                  d   j	                  |      }| j                  d   j	                  |      }| j                  d   j	                  |      } ||      5  |j                  ||||      }d d d        t        j                  |      S # 1 sw Y   xY w)Nr   )mpworkprecr%   r$   r.   )mpmathrK   rL   r'   
_to_mpmath	spherharmr   _from_mpmath)	r)   precrK   rL   r   r   r   r   ress	            r   _eval_evalfzYnm._eval_evalf   s     	(IIaL##D)IIaL##D)		!''-iil%%d+d^ 	1,,q!UC0C	1  d++	1 	1s   	B<<CN)r&   )T)__name__
__module____qualname____doc__classmethodr    r,   r0   r7   r;   rB   rD   rI   rS    r!   r   r   r      sE    wr 
; 
;=5&&
!(?
,r!   r   c                 0    t        t        | |||            S )a0  
    Conjugate spherical harmonics defined as

    .. math::
        \overline{Y_n^m(\theta, \varphi)} := (-1)^m Y_n^{-m}(\theta, \varphi).

    Examples
    ========

    >>> from sympy import Ynm_c, Symbol, simplify
    >>> from sympy.abc import n,m
    >>> theta = Symbol("theta")
    >>> phi = Symbol("phi")
    >>> Ynm_c(n, m, theta, phi)
    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
    >>> Ynm_c(n, m, -theta, phi)
    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

    For specific integers $n$ and $m$ we can evaluate the harmonics
    to more useful expressions:

    >>> simplify(Ynm_c(0, 0, theta, phi).expand(func=True))
    1/(2*sqrt(pi))
    >>> simplify(Ynm_c(1, -1, theta, phi).expand(func=True))
    sqrt(6)*exp(I*(-phi + 2*conjugate(phi)))*sin(theta)/(4*sqrt(pi))

    See Also
    ========

    Ynm, Znm

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/

    )r   r   )r   r   r   r   s       r   Ynm_cr[      s    P SAuc*++r!   c                        e Zd ZdZed        Zy)Znma{  
    Real spherical harmonics defined as

    .. math::

        Z_n^m(\theta, \varphi) :=
        \begin{cases}
          \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &\quad m > 0 \\
          Y_n^m(\theta, \varphi) &\quad m = 0 \\
          \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\
        \end{cases}

    which gives in simplified form

    .. math::

        Z_n^m(\theta, \varphi) =
        \begin{cases}
          \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\
          Y_n^m(\theta, \varphi) &\quad m = 0 \\
          \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &\quad m < 0 \\
        \end{cases}

    Examples
    ========

    >>> from sympy import Znm, Symbol, simplify
    >>> from sympy.abc import n, m
    >>> theta = Symbol("theta")
    >>> phi = Symbol("phi")
    >>> Znm(n, m, theta, phi)
    Znm(n, m, theta, phi)

    For specific integers n and m we can evaluate the harmonics
    to more useful expressions:

    >>> simplify(Znm(0, 0, theta, phi).expand(func=True))
    1/(2*sqrt(pi))
    >>> simplify(Znm(1, 1, theta, phi).expand(func=True))
    -sqrt(3)*sin(theta)*cos(phi)/(2*sqrt(pi))
    >>> simplify(Znm(2, 1, theta, phi).expand(func=True))
    -sqrt(15)*sin(2*theta)*cos(phi)/(4*sqrt(pi))

    See Also
    ========

    Ynm, Ynm_c

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/

    c                 "   |j                   r+t        ||||      t        ||||      z   t        d      z  }|S |j                  rt        ||||      S |j
                  r2t        ||||      t        ||||      z
  t        d      t        z  z  }|S y )Nr$   )is_positiver   r[   r   is_zerois_negativer   )r   r   r   r   r   zzs         r   r    zZnm.evalE  s    ==aE3'%1eS*AAT!WLBIYYq!UC((]]aE3'%1eS*AAd1gaiPBI r!   N)rT   rU   rV   rW   rX   r    rY   r!   r   r]   r]     s    7r  r!   r]   N)sympy.core.exprr   sympy.core.functionr   r   sympy.core.numbersr   r   sympy.core.singletonr   sympy.core.symbolr	   sympy.functionsr
   (sympy.functions.combinatorial.factorialsr   $sympy.functions.elementary.complexesr   r   &sympy.functions.elementary.exponentialr   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   r   _xr   r[   r]   rY   r!   r   <module>ro      sU      < $ " # * > ? 6 9 B B
3ZN,( N,b(,VC( Cr!   