
    "`                     V    d dl mZmZ edd       Zed        Zed        Zed        Zy)   )defundefun_wrappedNc                 T   	  j                         j                  n j                        dk  rt        d      n j                        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        	|r|r 	fd} j                  |      S  	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

        zn cannot be negativer   z#q-function only defined for |q| < 1maxterms2   c               3      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                   Gw)Nr      r      )NoConvergence)tkx1x2ctxr   qs       =/usr/lib/python3/dist-packages/mpmath/functions/qfunctions.pytermszqp.<locals>.termsf   s     AGABABAgl"Agl"a!A#a%j a!A#a%j Qx<+++ s   AAc               3      K   d} j                   }	 d|z  z
   |z  }| dz  } | k\  ry | kD  rj                  ,w)Nr   r   )oner   )r   rar   r   nr   s     r   factorszqp.<locals>.factorsv   sY     GGac'MFAFAAv8|''' s   <?)
convertinf
ValueErrorr   abszerogetprecsum_accuratelymul_accurately)
r   r   r   r   kwargsinfinitesamer   r   r   s
   ````     @r   qpr(      s*   T 	AAyGGKKN1u/00yKKNAvwwAaC  SWWHFDq6Q;bAFxx!|#BCC!V77Q;zz*bk2HD	, !!%((
( 
( g&&    c                     t        |      dkD  r'| j                  |d|z        ||dz
  |dz
  z  dz  z  z  S  | j                  ||dfi | | j                  ||z  |dfi |z  d|z
  d|z
  z  z  S )a  
    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      ?N)r   qgammar(   )r   zr   r%   s       r   r+   r+      s    D 1vzzz!AaC acAaC[_!555366!Q''q!tQ''(+,Q3!A#,7 7r)   c                     | j                  |      rL| j                  |      dkD  r8t        | j                  |            } | j                  |||fi |d|z
  |z  z  S  | j                  |dz   |fi |S )a  
    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   )isint_reintr(   r+   )r   r,   r   r%   r   s        r   qfacr1      sq    : yy|
Q
OsvvaA((AaC!8333::ac1'''r)   c                     D cg c]  } j                  |       c}D cg c]  } j                  |       c} j                         j                        t              }t              }	d|	z   |z
  |j                  dd j                  z         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     K   	j                   } |  d}d}d}	 D ]  }d||z  z
  }| |z  }  D ]  }d||z  z
  }|st        | |z  }  | z  } |d
z  |
z  z  z  }|z  }| d|z
  z  } |dz  }| |z   |kD  r	j                  nw)Nr   r   r   )r   r   r   )r   qkr   xr   pba_sb_sr   dr   r   r,   s          r   r   zqhyper.<locals>.terms  s     GG "HQ  "H$$Q	
 FA"q27""A!GB!b&MAFAa%K8|'''! s   BB	)r   lenr!   r"   r#   )r   r8   r9   r   r,   r%   r   r7   r   sr   r:   r   s   `````      @@r   qhyperr=      s    Z $'
'a3;;q>
'C#&
'a3;;q>
'CAAAACACA	!AAzz*bk2H( (. e$$? (
's
   CC)NN)	functionsr   r   r(   r+   r1   r=    r)   r   <module>r@      s]    +}' }'~ $7 $7L ( (B K% K%r)   