a
    RG5d؟                     @   s  d 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 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 m!Z!m"Z"m#Z#m$Z$ e	dZ%G dd deZ&G dd de&Z'dd Z(G dd de&Z)G dd de&Z*G dd de&Z+G dd deZ,G dd deZ-G d d! d!e&Z.G d"d# d#eZ/G d$d% d%e&Z0G d&d' d'e&Z1G d(d) d)e&Z2d*S )+z
This module mainly implements special orthogonal polynomials.

See also functions.combinatorial.numbers which contains some
combinatorial polynomials.

    )Rational)FunctionArgumentIndexError)S)Dummy)binomial	factorialRisingFactorial)re)exp)floor)sqrt)cossec)gamma)hyper)chebyshevt_polychebyshevu_polygegenbauer_polyhermite_polyjacobi_polylaguerre_polylegendre_polyxc                   @   s$   e Zd ZdZedd Zdd ZdS )OrthogonalPolynomialz+Base class for orthogonal polynomials.
    c                 C   s*   |j r&|dkr&| t|tt|S d S )Nr   )
is_integer_ortho_polyint_xsubsclsnr    r#   _/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/functions/special/polynomials.py_eval_at_order    s    z#OrthogonalPolynomial._eval_at_orderc                 C   s   |  | jd | jd  S )Nr      )funcargs	conjugate)selfr#   r#   r$   _eval_conjugate%   s    z$OrthogonalPolynomial._eval_conjugateN)__name__
__module____qualname____doc__classmethodr%   r+   r#   r#   r#   r$   r      s   
r   c                   @   s6   e Zd ZdZedd ZdddZdd Zd	d
 ZdS )jacobia  
    Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.

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

    ``jacobi(n, alpha, beta, x)`` gives the $n$th Jacobi polynomial
    in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$.

    The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
    to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.

    Examples
    ========

    >>> from sympy import jacobi, S, conjugate, diff
    >>> from sympy.abc import a, b, n, x

    >>> jacobi(0, a, b, x)
    1
    >>> jacobi(1, a, b, x)
    a/2 - b/2 + x*(a/2 + b/2 + 1)
    >>> jacobi(2, a, b, x)
    a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2

    >>> jacobi(n, a, b, x)
    jacobi(n, a, b, x)

    >>> jacobi(n, a, a, x)
    RisingFactorial(a + 1, n)*gegenbauer(n,
        a + 1/2, x)/RisingFactorial(2*a + 1, n)

    >>> jacobi(n, 0, 0, x)
    legendre(n, x)

    >>> jacobi(n, S(1)/2, S(1)/2, x)
    RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)

    >>> jacobi(n, -S(1)/2, -S(1)/2, x)
    RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)

    >>> jacobi(n, a, b, -x)
    (-1)**n*jacobi(n, b, a, x)

    >>> jacobi(n, a, b, 0)
    gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(2**n*factorial(n)*gamma(a + 1))
    >>> jacobi(n, a, b, 1)
    RisingFactorial(a + 1, n)/factorial(n)

    >>> conjugate(jacobi(n, a, b, x))
    jacobi(n, conjugate(a), conjugate(b), conjugate(x))

    >>> diff(jacobi(n,a,b,x), x)
    (a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)

    See Also
    ========

    gegenbauer,
    chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly,
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
    .. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/JacobiP/

    c                 C   s   ||kr|t ddkr4ttj|t| t|| S |jrDt||S |tjkrttdtj |t|d  t|| S t|d |td| d | t	||tj | S nV|| k rt
|| d t
|d  d| |d   d| |d   t|| | S |js| r*tj| t||||  S |jr~d|  t
|| d  t
|d t|  t| | | g|d gd S |tjkrt|d |t| S |tju r|jr|| d|  jrtdt|| | d |tj S nt||||S d S )N      r&   z,Error. a + b + 2*n should not be an integer.)r   r	   r   Halfr   
chebyshevtis_zerolegendre
chebyshevu
gegenbauerr   assoc_legendre	is_Numbercould_extract_minus_signNegativeOner1   r   OneInfinityis_positiver   
ValueErrorr   )r!   r"   abr   r#   r#   r$   eval~   s2    

&4J
,zjacobi.eval   c           
   	   C   s  ddl m} |dkr"t| |n|dkr| j\}}}}td}d|| | | d  }|| d|  d t|| d ||  || t|| | d ||   }	||t|||||	t||||   |d|d fS |dkr| j\}}}}td}d|| | | d  }d||  || d|  d t|| d ||  || t|| | d ||    }	||t|||||	t||||   |d|d fS |dkr| j\}}}}tj|| | d  t|d |d |d | S t| |d S )	Nr   Sumr&   r3   kr4   r2   rF   )	sympy.concrete.summationsrH   r   r(   r   r	   r1   r   r5   )
r*   argindexrH   r"   rC   rD   r   rI   f1f2r#   r#   r$   fdiff   s.    ( 4
2 4
0zjacobi.fdiffc           	      K   s   ddl m} |js|jdu r$tdtd}t| |t|| | d | t|| d ||  t| d| d |  }dt| |||d|f S )Nr   rG   F*Error: n should be a non-negative integer.rI   r&   r3   )rJ   rH   is_negativer   rB   r   r	   r   )	r*   r"   rC   rD   r   kwargsrH   rI   kernr#   r#   r$   _eval_rewrite_as_polynomial   s    6z"jacobi._eval_rewrite_as_polynomialc                 C   s*   | j \}}}}| || | | S Nr(   r'   r)   )r*   r"   rC   rD   r   r#   r#   r$   r+      s    zjacobi._eval_conjugateN)rF   	r,   r-   r.   r/   r0   rE   rN   rS   r+   r#   r#   r#   r$   r1   -   s   P
%

r1   c                 C   sz   t d|| d  t| | d t| | d   d|  | | d  t| t| | | d   }t| |||t| S )a  
    Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.

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

    ``jacobi_normalized(n, alpha, beta, x)`` gives the $n$th
    Jacobi polynomial in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$.

    The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
    to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.

    This functions returns the polynomials normilzed:

    .. math::

        \int_{-1}^{1}
          P_m^{\left(\alpha, \beta\right)}(x)
          P_n^{\left(\alpha, \beta\right)}(x)
          (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x
        = \delta_{m,n}

    Examples
    ========

    >>> from sympy import jacobi_normalized
    >>> from sympy.abc import n,a,b,x

    >>> jacobi_normalized(n, a, b, x)
    jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))

    Parameters
    ==========

    n : integer degree of polynomial

    a : alpha value

    b : beta value

    x : symbol

    See Also
    ========

    gegenbauer,
    chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly,
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
    .. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/JacobiP/

    r3   r&   )r   r   r   r1   r   )r"   rC   rD   r   Znfactorr#   r#   r$   jacobi_normalized   s    C2rW   c                   @   s6   e Zd ZdZedd ZdddZdd Zd	d
 ZdS )r:   a  
    Gegenbauer polynomial $C_n^{\left(\alpha\right)}(x)$.

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

    ``gegenbauer(n, alpha, x)`` gives the $n$th Gegenbauer polynomial
    in $x$, $C_n^{\left(\alpha\right)}(x)$.

    The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with
    respect to the weight $\left(1-x^2\right)^{\alpha-\frac{1}{2}}$.

    Examples
    ========

    >>> from sympy import gegenbauer, conjugate, diff
    >>> from sympy.abc import n,a,x
    >>> gegenbauer(0, a, x)
    1
    >>> gegenbauer(1, a, x)
    2*a*x
    >>> gegenbauer(2, a, x)
    -a + x**2*(2*a**2 + 2*a)
    >>> gegenbauer(3, a, x)
    x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)

    >>> gegenbauer(n, a, x)
    gegenbauer(n, a, x)
    >>> gegenbauer(n, a, -x)
    (-1)**n*gegenbauer(n, a, x)

    >>> gegenbauer(n, a, 0)
    2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))
    >>> gegenbauer(n, a, 1)
    gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))

    >>> conjugate(gegenbauer(n, a, x))
    gegenbauer(n, conjugate(a), conjugate(x))

    >>> diff(gegenbauer(n, a, x), x)
    2*a*gegenbauer(n - 1, a + 1, x)

    See Also
    ========

    jacobi,
    chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials
    .. [2] http://mathworld.wolfram.com/GegenbauerPolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/GegenbauerC3/

    c                 C   s  |j rtjS |tjkr t||S |tjkr4t||S |tjkrDtjS |js|tjkrt	|tjkdkrntj
S ttj||  ttj|  td| |  td| t|d   S | rtj| t|||  S |jr&d| ttj t|tj|   td| d t|d  t|  S |tjkrZtd| | td| t|d   S |tju r|jrt||tj S nt|||S d S )NTr3   r&   )rP   r   Zeror5   r8   r?   r9   r>   r<   r
   ComplexInfinityr   Pir   r   r=   r:   r7   r   r@   rA   r	   r   )r!   r"   rC   r   r#   r#   r$   rE   a  s6    





.""(zgegenbauer.evalr4   c           
      C   s&  ddl m} |dkr"t| |n |dkr| j\}}}td}ddd||    ||  || d|  ||   }d|d  |d|  d| d|  d   d|| d|    }|t||| |t|||  }	||	|d|d fS |dkr| j\}}}d| t|d |d | S t| |d S )Nr   rG   r&   r3   rI   r2   r4   )rJ   rH   r   r(   r   r:   )
r*   rK   rH   r"   rC   r   rI   Zfactor1Zfactor2rR   r#   r#   r$   rN     s,    * 
zgegenbauer.fdiffc                 K   sn   ddl m} td}d| t|||  d| |d|    t|t|d|    }|||dt|d fS )Nr   rG   rI   r2   r3   )rJ   rH   r   r	   r   r   )r*   r"   rC   r   rQ   rH   rI   rR   r#   r#   r$   rS     s    (z&gegenbauer._eval_rewrite_as_polynomialc                 C   s"   | j \}}}| || | S rT   rU   )r*   r"   rC   r   r#   r#   r$   r+     s    zgegenbauer._eval_conjugateN)r4   rV   r#   r#   r#   r$   r:     s   C
(
r:   c                   @   s6   e Zd ZdZeeZedd Zd
ddZ	dd Z
d	S )r6   a  
    Chebyshev polynomial of the first kind, $T_n(x)$.

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

    ``chebyshevt(n, x)`` gives the $n$th Chebyshev polynomial (of the first
    kind) in $x$, $T_n(x)$.

    The Chebyshev polynomials of the first kind are orthogonal on
    $[-1, 1]$ with respect to the weight $\frac{1}{\sqrt{1-x^2}}$.

    Examples
    ========

    >>> from sympy import chebyshevt, diff
    >>> from sympy.abc import n,x
    >>> chebyshevt(0, x)
    1
    >>> chebyshevt(1, x)
    x
    >>> chebyshevt(2, x)
    2*x**2 - 1

    >>> chebyshevt(n, x)
    chebyshevt(n, x)
    >>> chebyshevt(n, -x)
    (-1)**n*chebyshevt(n, x)
    >>> chebyshevt(-n, x)
    chebyshevt(n, x)

    >>> chebyshevt(n, 0)
    cos(pi*n/2)
    >>> chebyshevt(n, -1)
    (-1)**n

    >>> diff(chebyshevt(n, x), x)
    n*chebyshevu(n - 1, x)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
    .. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
    .. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
    .. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
    .. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/

    c                 C   s   |j st| r$tj| t||  S | r8t| |S |jrRttjtj | S |tj	krbtj	S |tj
u rtj
S n |jr| | |S | ||S d S rT   )r<   r=   r   r>   r6   r7   r   r5   rZ   r?   r@   rP   r%   r    r#   r#   r$   rE     s    

zchebyshevt.evalr3   c                 C   sF   |dkrt | |n.|dkr8| j\}}|t|d | S t | |d S Nr&   r3   )r   r(   r9   r*   rK   r"   r   r#   r#   r$   rN     s    
zchebyshevt.fdiffc                 K   sZ   ddl m} td}t|d| |d d |  ||d|    }|||dt|d fS Nr   rG   rI   r3   r&   )rJ   rH   r   r   r   r*   r"   r   rQ   rH   rI   rR   r#   r#   r$   rS     s    .z&chebyshevt._eval_rewrite_as_polynomialN)r3   )r,   r-   r.   r/   staticmethodr   r   r0   rE   rN   rS   r#   r#   r#   r$   r6     s   B

r6   c                   @   s6   e Zd ZdZeeZedd Zd
ddZ	dd Z
d	S )r9   a  
    Chebyshev polynomial of the second kind, $U_n(x)$.

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

    ``chebyshevu(n, x)`` gives the $n$th Chebyshev polynomial of the second
    kind in x, $U_n(x)$.

    The Chebyshev polynomials of the second kind are orthogonal on
    $[-1, 1]$ with respect to the weight $\sqrt{1-x^2}$.

    Examples
    ========

    >>> from sympy import chebyshevu, diff
    >>> from sympy.abc import n,x
    >>> chebyshevu(0, x)
    1
    >>> chebyshevu(1, x)
    2*x
    >>> chebyshevu(2, x)
    4*x**2 - 1

    >>> chebyshevu(n, x)
    chebyshevu(n, x)
    >>> chebyshevu(n, -x)
    (-1)**n*chebyshevu(n, x)
    >>> chebyshevu(-n, x)
    -chebyshevu(n - 2, x)

    >>> chebyshevu(n, 0)
    cos(pi*n/2)
    >>> chebyshevu(n, 1)
    n + 1

    >>> diff(chebyshevu(n, x), x)
    (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
    .. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
    .. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
    .. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
    .. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/

    c                 C   s   |j s| r$tj| t||  S | r\|tjkr<tjS | d  s\t| d | S |jrvttjtj	 | S |tj
krtj
| S |tju rtjS n8|jr|tjkrtjS | | d | S n| ||S d S )Nr3   )r<   r=   r   r>   r9   rX   r7   r   r5   rZ   r?   r@   rP   r%   r    r#   r#   r$   rE   e  s&    




zchebyshevu.evalr3   c                 C   sd   |dkrt | |nL|dkrV| j\}}|d t|d | |t||  |d d  S t | |d S r[   )r   r(   r6   r9   r\   r#   r#   r$   rN     s    
0zchebyshevu.fdiffc                 K   sn   ddl m} td}tj| t||  d| |d|    t|t|d|    }|||dt|d fS Nr   rG   rI   r3   rJ   rH   r   r   r>   r   r   r^   r#   r#   r$   rS     s    
z&chebyshevu._eval_rewrite_as_polynomialN)r3   )r,   r-   r.   r/   r_   r   r   r0   rE   rN   rS   r#   r#   r#   r$   r9      s   B

r9   c                   @   s   e Zd ZdZedd ZdS )chebyshevt_rootaR  
    ``chebyshev_root(n, k)`` returns the $k$th root (indexed from zero) of
    the $n$th Chebyshev polynomial of the first kind; that is, if
    $0 \le k < n$, ``chebyshevt(n, chebyshevt_root(n, k)) == 0``.

    Examples
    ========

    >>> from sympy import chebyshevt, chebyshevt_root
    >>> chebyshevt_root(3, 2)
    -sqrt(3)/2
    >>> chebyshevt(3, chebyshevt_root(3, 2))
    0

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly
    c                 C   s>   d|kr||k s t d||f ttjd| d  d|  S )Nr   +must have 0 <= k < n, got k = %s and n = %sr3   r&   rB   r   r   rZ   r!   r"   rI   r#   r#   r$   rE     s
    zchebyshevt_root.evalNr,   r-   r.   r/   r0   rE   r#   r#   r#   r$   rb     s   rb   c                   @   s   e Zd ZdZedd ZdS )chebyshevu_roota<  
    ``chebyshevu_root(n, k)`` returns the $k$th root (indexed from zero) of the
    $n$th Chebyshev polynomial of the second kind; that is, if $0 \le k < n$,
    ``chebyshevu(n, chebyshevu_root(n, k)) == 0``.

    Examples
    ========

    >>> from sympy import chebyshevu, chebyshevu_root
    >>> chebyshevu_root(3, 2)
    -sqrt(2)/2
    >>> chebyshevu(3, chebyshevu_root(3, 2))
    0

    See Also
    ========

    chebyshevt, chebyshevt_root, chebyshevu,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly
    c                 C   s:   d|kr||k s t d||f ttj|d  |d  S )Nr   rc   r&   rd   re   r#   r#   r$   rE     s
    zchebyshevu_root.evalNrf   r#   r#   r#   r$   rg     s   rg   c                   @   s6   e Zd ZdZeeZedd Zd
ddZ	dd Z
d	S )r8   ay  
    ``legendre(n, x)`` gives the $n$th Legendre polynomial of $x$, $P_n(x)$

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

    The Legendre polynomials are orthogonal on $[-1, 1]$ with respect to
    the constant weight 1. They satisfy $P_n(1) = 1$ for all $n$; further,
    $P_n$ is odd for odd $n$ and even for even $n$.

    Examples
    ========

    >>> from sympy import legendre, diff
    >>> from sympy.abc import x, n
    >>> legendre(0, x)
    1
    >>> legendre(1, x)
    x
    >>> legendre(2, x)
    3*x**2/2 - 1/2
    >>> legendre(n, x)
    legendre(n, x)
    >>> diff(legendre(n,x), x)
    n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Legendre_polynomial
    .. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/LegendreP/
    .. [4] http://functions.wolfram.com/Polynomials/LegendreP2/

    c                 C   s   |j s| r$tj| t||  S | rL| d  sLt| tj |S |jrttjt	tj
|d  t	tj|d    S |tjkrtjS |tju rtjS n|jr| tj }| ||S d S r[   )r<   r=   r   r>   r8   r?   r7   r   rZ   r   r5   r@   rP   r%   r    r#   r#   r$   rE   #  s    .

zlegendre.evalr3   c                 C   s`   |dkrt | |nH|dkrR| j\}}||d d  |t|| t|d |  S t | |d S r[   )r   r(   r8   r\   r#   r#   r$   rN   ;  s    
,zlegendre.fdiffc                 K   s`   ddl m} td}tj| t||d  d| d ||   d| d |  }|||d|fS r]   )rJ   rH   r   r   r>   r   r^   r#   r#   r$   rS   S  s    <z$legendre._eval_rewrite_as_polynomialN)r3   )r,   r-   r.   r/   r_   r   r   r0   rE   rN   rS   r#   r#   r#   r$   r8     s   4

r8   c                   @   sB   e Zd ZdZedd Zedd ZdddZd	d
 Zdd Z	dS )r;   a]  
    ``assoc_legendre(n, m, x)`` gives $P_n^m(x)$, where $n$ and $m$ are
    the degree and order or an expression which is related to the nth
    order Legendre polynomial, $P_n(x)$ in the following manner:

    .. math::
        P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}
                   \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}

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

    Associated Legendre polynomials are orthogonal on $[-1, 1]$ with:

    - weight $= 1$            for the same $m$ and different $n$.
    - weight $= \frac{1}{1-x^2}$   for the same $n$ and different $m$.

    Examples
    ========

    >>> from sympy import assoc_legendre
    >>> from sympy.abc import x, m, n
    >>> assoc_legendre(0,0, x)
    1
    >>> assoc_legendre(1,0, x)
    x
    >>> assoc_legendre(1,1, x)
    -sqrt(1 - x**2)
    >>> assoc_legendre(n,m,x)
    assoc_legendre(n, m, x)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
    .. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/LegendreP/
    .. [4] http://functions.wolfram.com/Polynomials/LegendreP2/

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r|j
r|jrtd| |f t||krtd| ||f | t|tt|t|S d S )Nr   r3   r&   z3%s : 1st index must be nonnegative integer (got %r)z9%s : abs('2nd index') must be <= '1st index' (got %r, %r))r=   r   r>   r   r;   r8   r   rZ   r   r<   r   rP   rB   absr%   r   r   r   )r!   r"   rk   r   r#   r#   r$   rE     s    2
:zassoc_legendre.evalr4   c                 C   s   |dkrt | |nn|dkr(t | |nZ|dkrx| j\}}}d|d d  || t||| || t|d ||   S t | |d S )Nr&   r3   r4   )r   r(   r;   )r*   rK   r"   rk   r   r#   r#   r$   rN     s    <zassoc_legendre.fdiffc                 K   s   ddl m} td}td| d|  d| t||  t| t|d|  |   tj|  ||| d|    }d|d  |d  |||dt|| tj f S r]   )rJ   rH   r   r   r   r>   r   r5   )r*   r"   rk   r   rQ   rH   rI   rR   r#   r#   r$   rS     s    &z*assoc_legendre._eval_rewrite_as_polynomialc                 C   s"   | j \}}}| || | S rT   rU   )r*   r"   rk   r   r#   r#   r$   r+     s    zassoc_legendre._eval_conjugateN)r4   )
r,   r-   r.   r/   r0   r%   rE   rN   rS   r+   r#   r#   r#   r$   r;   Z  s   9


r;   c                   @   s6   e Zd ZdZeeZedd Zd
ddZ	dd Z
d	S )hermitea  
    ``hermite(n, x)`` gives the $n$th Hermite polynomial in $x$, $H_n(x)$

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

    The Hermite polynomials are orthogonal on $(-\infty, \infty)$
    with respect to the weight $\exp\left(-x^2\right)$.

    Examples
    ========

    >>> from sympy import hermite, diff
    >>> from sympy.abc import x, n
    >>> hermite(0, x)
    1
    >>> hermite(1, x)
    2*x
    >>> hermite(2, x)
    4*x**2 - 2
    >>> hermite(n, x)
    hermite(n, x)
    >>> diff(hermite(n,x), x)
    2*n*hermite(n - 1, x)
    >>> hermite(n, -x)
    (-1)**n*hermite(n, x)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
    .. [2] http://mathworld.wolfram.com/HermitePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/HermiteH/

    c                 C   s   |j s`| r$tj| t||  S |jrNd| ttj ttj	| d  S |tj
u rtj
S n |jrttd| n| ||S d S )Nr3   0The index n must be nonnegative integer (got %r))r<   r=   r   r>   rn   r7   r   rZ   r   r?   r@   rP   rB   r%   r    r#   r#   r$   rE     s    $
zhermite.evalr3   c                 C   sJ   |dkrt | |n2|dkr<| j\}}d| t|d | S t | |d S r[   )r   r(   rn   r\   r#   r#   r$   rN     s    
zhermite.fdiffc                 K   sj   ddl m} td}tj| t|t|d|    d| |d|    }t||||dt|d f S r`   ra   r^   r#   r#   r$   rS     s    6z#hermite._eval_rewrite_as_polynomialN)r3   )r,   r-   r.   r/   r_   r   r   r0   rE   rN   rS   r#   r#   r#   r$   rn     s   3

rn   c                   @   s6   e Zd ZdZeeZedd Zd
ddZ	dd Z
d	S )laguerrea  
    Returns the $n$th Laguerre polynomial in $x$, $L_n(x)$.

    Examples
    ========

    >>> from sympy import laguerre, diff
    >>> from sympy.abc import x, n
    >>> laguerre(0, x)
    1
    >>> laguerre(1, x)
    1 - x
    >>> laguerre(2, x)
    x**2/2 - 2*x + 1
    >>> laguerre(3, x)
    -x**3/6 + 3*x**2/2 - 3*x + 1

    >>> laguerre(n, x)
    laguerre(n, x)

    >>> diff(laguerre(n, x), x)
    -assoc_laguerre(n - 1, 1, x)

    Parameters
    ==========

    n : int
        Degree of Laguerre polynomial. Must be `n \ge 0`.

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial
    .. [2] http://mathworld.wolfram.com/LaguerrePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
    .. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/

    c                 C   s   |j du rtd|js| rH| d  sHt|t| d |  S |jrTtjS |tj	u rdtj
S |tj
u rtj| tj
 S n,|jrt|t| d |  S | ||S d S )NFError: n should be an integer.r&   )r   rB   r<   r=   r   rp   r7   r   r?   NegativeInfinityr@   r>   rP   r%   r    r#   r#   r$   rE   c  s    


zlaguerre.evalr3   c                 C   sF   |dkrt | |n.|dkr8| j\}}t|d d| S t | |d S r[   )r   r(   assoc_laguerrer\   r#   r#   r$   rN   z  s    
zlaguerre.fdiffc                 K   s   ddl m} |jr6t|| j| d | fi | S |jdu rHtdtd}t| |t	|d  ||  }|||d|fS )Nr   rG   r&   Frq   rI   r3   )
rJ   rH   rP   r   rS   r   rB   r   r	   r   r^   r#   r#   r$   rS     s    $
 z$laguerre._eval_rewrite_as_polynomialN)r3   )r,   r-   r.   r/   r_   r   r   r0   rE   rN   rS   r#   r#   r#   r$   rp   )  s   7

rp   c                   @   s6   e Zd ZdZedd ZdddZdd Zd	d
 ZdS )rs   a,  
    Returns the $n$th generalized Laguerre polynomial in $x$, $L_n(x)$.

    Examples
    ========

    >>> from sympy import assoc_laguerre, diff
    >>> from sympy.abc import x, n, a
    >>> assoc_laguerre(0, a, x)
    1
    >>> assoc_laguerre(1, a, x)
    a - x + 1
    >>> assoc_laguerre(2, a, x)
    a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1
    >>> assoc_laguerre(3, a, x)
    a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +
        x*(-a**2/2 - 5*a/2 - 3) + 1

    >>> assoc_laguerre(n, a, 0)
    binomial(a + n, a)

    >>> assoc_laguerre(n, a, x)
    assoc_laguerre(n, a, x)

    >>> assoc_laguerre(n, 0, x)
    laguerre(n, x)

    >>> diff(assoc_laguerre(n, a, x), x)
    -assoc_laguerre(n - 1, a + 1, x)

    >>> diff(assoc_laguerre(n, a, x), a)
    Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))

    Parameters
    ==========

    n : int
        Degree of Laguerre polynomial. Must be `n \ge 0`.

    alpha : Expr
        Arbitrary expression. For ``alpha=0`` regular Laguerre
        polynomials will be generated.

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials
    .. [2] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
    .. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/

    c                 C   s   |j rt||S |jsf|j r*t|| |S |tju rL|dkrLtj| tj S |tju r|dkrtjS n |jrzt	d| nt
|||S d S )Nr   ro   )r7   rp   r<   r   r   r@   r>   rr   rP   rB   r   )r!   r"   alphar   r#   r#   r$   rE     s    
zassoc_laguerre.evalr4   c                 C   s   ddl m} |dkr t| |nt|dkr`| j\}}}td}|t|||||  |d|d fS |dkr| j\}}}t|d |d | S t| |d S )Nr   rG   r&   r3   rI   r4   )rJ   rH   r   r(   r   rs   )r*   rK   rH   r"   rt   r   rI   r#   r#   r$   rN     s    $zassoc_laguerre.fdiffc                 K   s   ddl m} |js|jdu r$tdtd}t| |t|| d t|  ||  }t|| d t| |||d|f S )Nr   rG   FrO   rI   r&   )	rJ   rH   rP   r   rB   r   r	   r   r   )r*   r"   rt   r   rQ   rH   rI   rR   r#   r#   r$   rS     s    z*assoc_laguerre._eval_rewrite_as_polynomialc                 C   s"   | j \}}}| || | S rT   rU   )r*   r"   rt   r   r#   r#   r$   r+     s    zassoc_laguerre._eval_conjugateN)r4   rV   r#   r#   r#   r$   rs     s   E


rs   N)3r/   
sympy.corer   sympy.core.functionr   r   sympy.core.singletonr   sympy.core.symbolr   (sympy.functions.combinatorial.factorialsr   r   r	   $sympy.functions.elementary.complexesr
   &sympy.functions.elementary.exponentialr   #sympy.functions.elementary.integersr   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   'sympy.functions.special.gamma_functionsr   sympy.functions.special.hyperr   sympy.polys.orthopolysr   r   r   r   r   r   r   r   r   r1   rW   r:   r6   r9   rb   rg   r8   r;   rn   rp   rs   r#   r#   r#   r$   <module>   s:   $ #N px(,no`h