Ë
    3^(hÑ  ã                   óV   — d dl mZmZ edd„«       Zed„ «       Zed„ «       Zed„ «       Zy)é   )ÚdefunÚdefun_wrappedNc                 óT  ‡ ‡‡‡‡	— ‰ j                  ‰«      Š‰€‰ j                  Šn‰ j                  ‰«      Š‰dk  rt        d«      ‚‰€‰Šn‰ j                  ‰«      Š‰dk(  r‰ j                  d‰‰z   z  z   S ‰‰ j                  k(  }‰‰k(  }|rHt	        ‰«      dk\  r&|r‰dk(  s‰dk(  r‰ j
                  ‰z  S t        d«      ‚‰dk(  r‰ j                  ‰z
  S |j                  dd‰ j                  z  «      Š	|r|rˆ ˆ	ˆfd„}‰ j                  |«      S ˆˆ ˆ	ˆˆfd	„}‰ j                  |«      S )
a  
    Evaluates the q-Pochhammer symbol (or q-rising factorial)

    .. math ::

        (a; q)_n = \prod_{k=0}^{n-1} (1-a q^k)

    where `n = \infty` is permitted if `|q| < 1`. Called with two arguments,
    ``qp(a,q)`` computes `(a;q)_{\infty}`; with a single argument, ``qp(q)``
    computes `(q;q)_{\infty}`. The special case

    .. math ::

        \phi(q) = (q; q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k) =
            \sum_{k=-\infty}^{\infty} (-1)^k q^{(3k^2-k)/2}

    is also known as the Euler function, or (up to a factor `q^{-1/24}`)
    the Dedekind eta function.

    **Examples**

    If `n` is a positive integer, the function amounts to a finite product::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> qp(2,3,5)
        -725305.0
        >>> fprod(1-2*3**k for k in range(5))
        -725305.0
        >>> qp(2,3,0)
        1.0

    Complex arguments are allowed::

        >>> qp(2-1j, 0.75j)
        (0.4628842231660149089976379 + 4.481821753552703090628793j)

    The regular Pochhammer symbol `(a)_n` is obtained in the
    following limit as `q \to 1`::

        >>> a, n = 4, 7
        >>> limit(lambda q: qp(q**a,q,n) / (1-q)**n, 1)
        604800.0
        >>> rf(a,n)
        604800.0

    The Taylor series of the reciprocal Euler function gives
    the partition function `P(n)`, i.e. the number of ways of writing
    `n` as a sum of positive integers::

        >>> taylor(lambda q: 1/qp(q), 0, 10)
        [1.0, 1.0, 2.0, 3.0, 5.0, 7.0, 11.0, 15.0, 22.0, 30.0, 42.0]

    Special values include::

        >>> qp(0)
        1.0
        >>> findroot(diffun(qp), -0.4)   # location of maximum
        -0.4112484791779547734440257
        >>> qp(_)
        1.228348867038575112586878

    The q-Pochhammer symbol is related to the Jacobi theta functions.
    For example, the following identity holds::

        >>> q = mpf(0.5)    # arbitrary
        >>> qp(q)
        0.2887880950866024212788997
        >>> root(3,-2)*root(q,-24)*jtheta(2,pi/6,root(q,6))
        0.2887880950866024212788997

    é    zn cannot be negativer   éÿÿÿÿz#q-function only defined for |q| < 1Úmaxtermsé2   c               3   óº   •K  — d} | –— d}‰}‰dz  }	 d|z  |z  –— d|z  |z  –— |‰d|z  dz   z  z  }|‰d|z  dz   z  z  }|dz  }|‰kD  r‰j                   ‚ŒG­w)Nr   é   r   é   )ÚNoConvergence)ÚtÚkÚx1Úx2Úctxr   Úqs       €€€úY/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/mpmath/functions/qfunctions.pyÚtermszqp.<locals>.termsf   s�   øè ø€ ØˆAØŠGØˆAØˆBØ�A‘ˆBØØ˜A‘g ‘lÒ"Ø˜A‘g ‘lÒ"Ø�a˜!˜A™#˜a™%‘jÑ �Ø�a˜!˜A™#˜a™%‘jÑ �Ø�Q‘�Ø�x’<Ø×+Ñ+Ð+ð ùs   ƒAAc               3   ó‚   •K  — d} ‰j                   }	 d‰|z  z
  –— |‰z  }| dz  } | ‰k\  ry | ‰kD  r‰j                  ‚Œ,­w)Nr   r   )Úoner   )r   ÚrÚar   r   Únr   s     €€€€€r   Úfactorszqp.<locals>.factorsv   sY   øè ø€ ØˆØ�G‰GˆØØ�a˜‘c‘'ŠMØ�‰FˆAØ�‰FˆAØ�AŠvØØ�8Š|Ø×'Ñ'Ð'ð ùs   ƒ<?)
ÚconvertÚinfÚ
ValueErrorr   ÚabsÚzeroÚgetÚprecÚsum_accuratelyÚmul_accurately)
r   r   r   r   ÚkwargsÚinfiniteÚsamer   r   r   s
   ````     @r   Úqpr(      s*  ü€ ðT 	�‰�A‹€AØ€yØ�G‰G‰à�K‰K˜‹NˆØˆ1‚uÜÐ/Ó0Ð0Ø€yØ‰à�K‰K˜‹NˆØˆA‚vØ�w‰w˜˜A˜a™C™Ñ Ð Ø�S—W‘W‘€HØ�‰F€DÙÜˆq‹6�QŠ;Ù˜˜bš A¨¢FØ—x‘x !‘|Ð#ÜÐBÓCÐCØ�!ŠVØ—7‘7˜Q‘;ÐØ�z‰z˜* b¨¯©¡kÓ2€HÙ‘Dö	,ð ×!Ñ! %Ó(Ð(÷
(ð 
(ð ×Ñ˜gÓ&Ð&ó    c                 óà   — t        |«      dkD  r'| j                  |d|z  «      ||dz
  |dz
  z  dz  z  z  S  | j                  ||dfi |¤Ž | j                  ||z  |dfi |¤Žz  d|z
  d|z
  z  z  S )aþ  
    Evaluates the q-gamma function

    .. math ::

        \Gamma_q(z) = \frac{(q; q)_{\infty}}{(q^z; q)_{\infty}} (1-q)^{1-z}.


    **Examples**

    Evaluation for real and complex arguments::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> qgamma(4,0.75)
        4.046875
        >>> qgamma(6,6)
        121226245.0
        >>> qgamma(3+4j, 0.5j)
        (0.1663082382255199834630088 + 0.01952474576025952984418217j)

    The q-gamma function satisfies a functional equation similar
    to that of the ordinary gamma function::

        >>> q = mpf(0.25)
        >>> z = mpf(2.5)
        >>> qgamma(z+1,q)
        1.428277424823760954685912
        >>> (1-q**z)/(1-q)*qgamma(z,q)
        1.428277424823760954685912

    r   r   g      à?N)r   Úqgammar(   )r   Úzr   r%   s       r   r+   r+   ƒ   s�   € ôD ˆ1ƒv�‚zØ�z‰z˜!˜A˜a™CÓ   a¨¡c¨A¨a©C¡[°¡_Ñ!5Ñ5Ð5Øˆ3�6‰6�!�Q˜Ñ' Ñ'Øˆ�‰ˆq�!‰t�Q˜Ñ' Ñ'ñ(Ø+,¨Q©3°!°A±#©,ñ7ð 7r)   c                 óê   — | j                  |«      rL| j                  |«      dkD  r8t        | j                  |«      «      } | j                  |||fi |¤Žd|z
  |z  z  S  | j                  |dz   |fi |¤ŽS )aó  
    Evaluates the q-factorial,

    .. math ::

        [n]_q! = (1+q)(1+q+q^2)\cdots(1+q+\cdots+q^{n-1})

    or more generally

    .. math ::

        [z]_q! = \frac{(q;q)_z}{(1-q)^z}.

    **Examples**

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> qfac(0,0)
        1.0
        >>> qfac(4,3)
        2080.0
        >>> qfac(5,6)
        121226245.0
        >>> qfac(1+1j, 2+1j)
        (0.4370556551322672478613695 + 0.2609739839216039203708921j)

    r   r   )ÚisintÚ_reÚintr(   r+   )r   r,   r   r%   r   s        r   Úqfacr1   ª   sq   € ð: ‡y�y�„|˜Ÿ™ ›
 QšÜ�—‘˜“
‹OˆØˆs�v‰v�a˜˜AÑ( Ñ(¨A¨a©C°!©8Ñ3Ð3Øˆ3�:‰:�a˜‘c˜1Ñ' Ñ'Ð'r)   c                 ó–  ‡ ‡‡‡‡‡‡— ‰D �cg c]  }‰ j                  |«      ‘Œ c}Š‰D �cg c]  }‰ j                  |«      ‘Œ c}Š‰ j                  ‰«      Š‰ j                  ‰«      Št        ‰«      }t        ‰«      }	d|	z   |z
  Š|j                  dd‰ j                  z  «      Šˆˆˆ ˆˆˆˆfd„}
‰ j	                  |
«      S c c}w c c}w )a  
    Evaluates the basic hypergeometric series or hypergeometric q-series

    .. math ::

        \,_r\phi_s \left[\begin{matrix}
            a_1 & a_2 & \ldots & a_r \\
            b_1 & b_2 & \ldots & b_s
        \end{matrix} ; q,z \right] =
        \sum_{n=0}^\infty
        \frac{(a_1;q)_n, \ldots, (a_r;q)_n}
             {(b_1;q)_n, \ldots, (b_s;q)_n}
        \left((-1)^n q^{n\choose 2}\right)^{1+s-r}
        \frac{z^n}{(q;q)_n}

    where `(a;q)_n` denotes the q-Pochhammer symbol (see :func:`~mpmath.qp`).

    **Examples**

    Evaluation works for real and complex arguments::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> qhyper([0.5], [2.25], 0.25, 4)
        -0.1975849091263356009534385
        >>> qhyper([0.5], [2.25], 0.25-0.25j, 4)
        (2.806330244925716649839237 + 3.568997623337943121769938j)
        >>> qhyper([1+j], [2,3+0.5j], 0.25, 3+4j)
        (9.112885171773400017270226 - 1.272756997166375050700388j)

    Comparing with a summation of the defining series, using
    :func:`~mpmath.nsum`::

        >>> b, q, z = 3, 0.25, 0.5
        >>> qhyper([], [b], q, z)
        0.6221136748254495583228324
        >>> nsum(lambda n: z**n / qp(q,q,n)/qp(b,q,n) * q**(n*(n-1)), [0,inf])
        0.6221136748254495583228324

    r   r   r	   c               3   ó  •K  — ‰	j                   } | –— d}d}d}	 ‰D ]  }d||z  z
  }| |z  } Œ ‰D ]  }d||z  z
  }|st        ‚| |z  } Œ | ‰z  } |d‰
z  |‰
z  z  z  }|‰z  }| d|z
  z  } |dz  }| |z  –— |‰kD  r‰	j                  ‚Œn­w)Nr   r   r   )r   r   r   )r   Úqkr   Úxr   ÚpÚbÚa_sÚb_sr   Údr   r   r,   s          €€€€€€€r   r   zqhyper.<locals>.terms  sÞ   øè ø€ Ø�G‰GˆØŠØˆØˆØˆØØò �Ø˜˜"™‘H�Ø�Q‘‘ðð ò �Ø˜˜"™‘H�ÙÜ$Ð$Ø�Q‘‘ð	ð
 �‰FˆAØ�"�q‘˜2 ™7Ñ"Ñ"ˆAØ�!‰GˆBØ�!�b‘&‰MˆAØ�‰FˆAØ�a‘%ŠKØ�8Š|Ø×'Ñ'Ð'ð! ùs   ƒBB	)r   Úlenr!   r"   r#   )r   r8   r9   r   r,   r%   r   r7   r   Úsr   r:   r   s   `````      @@r   Úqhyperr=   Ì   s¬   þ€ ðZ $'Ö
'˜aˆ3�;‰;�q�>Ò
'€CØ#&Ö
'˜aˆ3�;‰;�q�>Ò
'€CØ�‰�A‹€AØ�‰�A‹€AÜˆC‹€AÜˆC‹€AØ	ˆ!‰ˆA‰€AØ�z‰z˜* b¨¯©¡kÓ2€H÷(ò (ð. ×Ñ˜eÓ$Ð$ùò? (ùÚ
's
   ŒC«C)NN)Ú	functionsr   r   r(   r+   r1   r=   © r)   r   ú<module>r@      s]   ðß +àò}'ó ð}'ð~ ñ$7ó ð$7ðL ñ(ó ð(ðB ñK%ó ñK%r)   