ó c·e[c@s\ddlmZmZeddd„ƒZed„ƒZed„ƒZed„ƒZdS(i(tdefunt defun_wrappedc  sˆˆjˆƒ‰ˆd kr'ˆj‰nˆjˆƒ‰ˆdkrQtdƒ‚nˆd krfˆ‰nˆjˆƒ‰ˆdkr”ˆjdˆˆSˆˆjk}ˆˆk}|rtˆƒdkrÿ|rðˆdksåˆdkrðˆjˆStdƒ‚qˆdkrˆjˆSn|jddˆjƒ‰|r`|r`‡‡‡fd†}ˆj |ƒS‡‡‡‡‡fd †}ˆj |ƒS( s 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 isn cannot be negativeiiÿÿÿÿs#q-function only defined for |q| < 1tmaxtermsi2c3s”d}|Vd}ˆ}ˆd}xld||Vd||V|ˆd|d9}|ˆd|d9}|d7}|ˆkr$ˆj‚q$q$WdS(Niiiÿÿÿÿi(t NoConvergence(tttktx1tx2(tctxRtq(s:lib/python2.7/site-packages/mpmath/functions/qfunctions.pyttermsfs     c3scd}ˆj}xMdˆ|V|ˆ9}|d7}|ˆkrCdS|ˆkrˆj‚qqWdS(Nii(toneR(Rtr(taRRtnR (s:lib/python2.7/site-packages/mpmath/functions/qfunctions.pytfactorsvs      N( tconverttNonetinft ValueErrorR tabstzerotgettprectsum_accuratelytmul_accurately( RR R RtkwargstinfinitetsameR R((R RRRR s:lib/python2.7/site-packages/mpmath/functions/qfunctions.pytqps4J            cKs~t|ƒdkr>|j|d|ƒ||d|ddS|j||d||j|||d|d|d|S(sþ Evaluates the q-gamma function .. math :: \Gamma_q(z) = \frac{(q; q)_{\infty}}{(q^z; q)_{\infty}} (1-q)^{1-z}. **Examples** Evaluation for real and complex arguments:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> qgamma(4,0.75) 4.046875 >>> qgamma(6,6) 121226245.0 >>> qgamma(3+4j, 0.5j) (0.1663082382255199834630088 + 0.01952474576025952984418217j) The q-gamma function satisfies a functional equation similar to that of the ordinary gamma function:: >>> q = mpf(0.25) >>> z = mpf(2.5) >>> qgamma(z+1,q) 1.428277424823760954685912 >>> (1-q**z)/(1-q)*qgamma(z,q) 1.428277424823760954685912 iigà?N(RtqgammaRR(RtzR R((s:lib/python2.7/site-packages/mpmath/functions/qfunctions.pyRƒs",cKsr|j|ƒr[|j|ƒdkr[t|j|ƒƒ}|j||||d||S|j|d||S(só Evaluates the q-factorial, .. math :: [n]_q! = (1+q)(1+q+q^2)\cdots(1+q+\cdots+q^{n-1}) or more generally .. math :: [z]_q! = \frac{(q;q)_z}{(1-q)^z}. **Examples** >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> qfac(0,0) 1.0 >>> qfac(4,3) 2080.0 >>> qfac(5,6) 121226245.0 >>> qfac(1+1j, 2+1j) (0.4370556551322672478613695 + 0.2609739839216039203708921j) ii(tisintt_retintRR(RRR RR((s:lib/python2.7/site-packages/mpmath/functions/qfunctions.pytqfacªs$"c  sÏgˆD]}ˆj|ƒ^q‰gˆD]}ˆj|ƒ^q)‰ˆjˆƒ‰ˆjˆƒ‰tˆƒ}tˆƒ} d| |‰|jddˆjƒ‰‡‡‡‡‡‡‡fd†} ˆj| ƒS(s 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 iRi2c3sïˆj}|Vd}d}d}xÈx&ˆD]}d||}||9}q*Wx5ˆD]-}d||}|svt‚n||:}qSW|ˆ9}|dˆ|ˆ9}|ˆ9}|d|:}|d7}||V|ˆkr#ˆj‚q#q#WdS(Niiiÿÿÿÿ(R RR(RtqkRtxR tptb(ta_stb_sRtdRR R(s:lib/python2.7/site-packages/mpmath/functions/qfunctions.pyR s,         (RtlenRRR( RR(R)R RRR R'R tsR ((R(R)RR*RR Rs:lib/python2.7/site-packages/mpmath/functions/qfunctions.pytqhyperÌs-""  !N(t functionsRRRRRR#R-(((s:lib/python2.7/site-packages/mpmath/functions/qfunctions.pyts '"