ó ¡¼™\c@ sdZddlmZmZddlmZmZmZmZm Z ddl m Z ddl m Z ddlmZmZmZddlmZmZddlmZd efd „ƒYZd efd „ƒYZd efd„ƒYZdefd„ƒYZdefd„ƒYZdS(s$ Riemann zeta and related function. iÿÿÿÿ(tprint_functiontdivision(tFunctiontStsympifytpitI(trange(tArgumentIndexError(t bernoullit factorialtharmonic(tlogt exp_polar(tsqrttlerchphicB s>eZdZd„Zdd„Zd„Zd„Zd„ZRS(s© Lerch transcendent (Lerch phi function). For :math:`\operatorname{Re}(a) > 0`, `|z| < 1` and `s \in \mathbb{C}`, the Lerch transcendent is defined as .. math :: \Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s}, where the standard branch of the argument is used for :math:`n + a`, and by analytic continuation for other values of the parameters. A commonly used related function is the Lerch zeta function, defined by .. math:: L(q, s, a) = \Phi(e^{2\pi i q}, s, a). **Analytic Continuation and Branching Behavior** It can be shown that .. math:: \Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}. This provides the analytic continuation to `\operatorname{Re}(a) \le 0`. Assume now `\operatorname{Re}(a) > 0`. The integral representation .. math:: \Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}} \frac{\mathrm{d}t}{\Gamma(s)} provides an analytic continuation to :math:`\mathbb{C} - [1, \infty)`. Finally, for :math:`x \in (1, \infty)` we find .. math:: \lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a) -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a) = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)}, using the standard branch for both :math:`\log{x}` and :math:`\log{\log{x}}` (a branch of :math:`\log{\log{x}}` is needed to evaluate :math:`\log{x}^{s-1}`). This concludes the analytic continuation. The Lerch transcendent is thus branched at :math:`z \in \{0, 1, \infty\}` and :math:`a \in \mathbb{Z}_{\le 0}`. For fixed :math:`z, a` outside these branch points, it is an entire function of :math:`s`. See Also ======== polylog, zeta References ========== .. [1] Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions, Vol. I, New York: McGraw-Hill. Section 1.11. .. [2] http://dlmf.nist.gov/25.14 .. [3] https://en.wikipedia.org/wiki/Lerch_transcendent Examples ======== The Lerch transcendent is a fairly general function, for this reason it does not automatically evaluate to simpler functions. Use expand_func() to achieve this. If :math:`z=1`, the Lerch transcendent reduces to the Hurwitz zeta function: >>> from sympy import lerchphi, expand_func >>> from sympy.abc import z, s, a >>> expand_func(lerchphi(1, s, a)) zeta(s, a) More generally, if :math:`z` is a root of unity, the Lerch transcendent reduces to a sum of Hurwitz zeta functions: >>> expand_func(lerchphi(-1, s, a)) 2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, a/2 + 1/2) If :math:`a=1`, the Lerch transcendent reduces to the polylogarithm: >>> expand_func(lerchphi(z, s, 1)) polylog(s, z)/z More generally, if :math:`a` is rational, the Lerch transcendent reduces to a sum of polylogarithms: >>> from sympy import S >>> expand_func(lerchphi(z, s, S(1)/2)) 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) - polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z)) >>> expand_func(lerchphi(z, s, S(3)/2)) -2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) - polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z The derivatives with respect to :math:`z` and :math:`a` can be computed in closed form: >>> lerchphi(z, s, a).diff(z) (-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z >>> lerchphi(z, s, a).diff(a) -s*lerchphi(z, s + 1, a) c K sddlm}m}m}m}m}m}m}m} |j \} } } | dkret | | ƒS| j r| dkr|dƒ} || | | | ƒ}dd| }t dƒ}x;t |jƒƒD]'}|||7}| |j| ƒ}qËW|j| | ƒS| jrÊt dƒ}t dƒ}| dkrª|| ƒ}|| krX|d8}n| |8} | | }|gt|ƒD]!}| || | || ^q}Œ}nu| dkr|| ƒd}| |7} | |}|gt|ƒD](}| |d|| |d| ^qëŒ}nt | j| jgƒ\}}|dt||ƒ}| d|}|||| d|gt|ƒD]<}t| |||ƒj|| |ƒ|||^q…ŒSt| |ƒrñ| j dt|js| d|| gkr| dkr.t ddgƒ\}}n†| |krUt ddgƒ\}}n_| | kr}t ddgƒ\}}n7| j ddt|}t |j|jgƒ\}}|gt|ƒD]C}|dt||||ƒ|| t | || |ƒ^qÄŒSt| | | ƒS(Niÿÿÿÿ(texpRtfloortAddtPolytDummyR t unpolarifyiittii(tsympyRRRRRRR Rtargstzetat is_IntegerRtreversedt all_coeffstdifftsubst is_RationalRtptqRtpolylogt_eval_expand_funct isinstanceR(tselfthintsRRRRRRR RtztstaRR tstarttrestctaddtmultntktmtzettrootR!targ((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyR#vs\:             =   DO=   TicC sy|j\}}}|dkr7| t||d|ƒS|dkrot||d|ƒ|t|||ƒ|St‚dS(Nii(RRR(R%targindexR'R(R)((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pytfdiff³s   ,cC s'|jƒ}|j|ƒr|S|SdS(N(R#thas(R%R'R(R)ttargetR+((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyt_eval_rewrite_helper¼s cK s|j|||tƒS(N(R9R(R%R'R(R)tkwargs((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyt_eval_rewrite_as_zetaÃscK s|j|||tƒS(N(R9R"(R%R'R(R)R:((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyt_eval_rewrite_as_polylogÆs(t__name__t __module__t__doc__R#R6R9R;R<(((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyRs d =  R"cB s;eZdZed„ƒZdd„Zd„Zd„ZRS(s/ Polylogarithm function. For :math:`|z| < 1` and :math:`s \in \mathbb{C}`, the polylogarithm is defined by .. math:: \operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s}, where the standard branch of the argument is used for :math:`n`. It admits an analytic continuation which is branched at :math:`z=1` (notably not on the sheet of initial definition), :math:`z=0` and :math:`z=\infty`. The name polylogarithm comes from the fact that for :math:`s=1`, the polylogarithm is related to the ordinary logarithm (see examples), and that .. math:: \operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t. The polylogarithm is a special case of the Lerch transcendent: .. math:: \operatorname{Li}_{s}(z) = z \Phi(z, s, 1) See Also ======== zeta, lerchphi Examples ======== For :math:`z \in \{0, 1, -1\}`, the polylogarithm is automatically expressed using other functions: >>> from sympy import polylog >>> from sympy.abc import s >>> polylog(s, 0) 0 >>> polylog(s, 1) zeta(s) >>> polylog(s, -1) -dirichlet_eta(s) If :math:`s` is a negative integer, :math:`0` or :math:`1`, the polylogarithm can be expressed using elementary functions. This can be done using expand_func(): >>> from sympy import expand_func >>> from sympy.abc import z >>> expand_func(polylog(1, z)) -log(1 - z) >>> expand_func(polylog(0, z)) z/(1 - z) The derivative with respect to :math:`z` can be computed in closed form: >>> polylog(s, z).diff(z) polylog(s - 1, z)/z The polylogarithm can be expressed in terms of the lerch transcendent: >>> from sympy import lerchphi >>> polylog(s, z).rewrite(lerchphi) z*lerchphi(z, s, 1) cC sbt||fƒ\}}|dkr.t|ƒS|dkrEt|ƒ S|dkrXtjS|dkrÎ|tjkr‘tddtdƒddS|dkr»tddtttdƒS|t dƒd dkrtd dtt dƒddƒddS|t dƒd dkrGtd d tt dƒddƒdS|d t dƒdkr‰tddtt dƒddƒdS|t dƒddkrtdd tt dƒddƒdSn4|dkræ|d|S|dkr|d|dSdd l m }m }m }|jt|ƒr^||ƒtjktkr^||||ƒƒSdS( Niiÿÿÿÿiii iiii i(tAbsRt polar_lift(RRt dirichlet_etaRtZerotHalfRR RRt$sympy.functions.elementary.complexesR@RRAR7R tOnetTrue(tclsR(R'R@RRA((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pytevals6       -)(+   -icC s:|j\}}|dkr0t|d|ƒ|St‚dS(Nii(RR"R(R%R5R(R'((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyR63s cK s|t||dƒS(Ni(R(R%R(R'R:((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyt_eval_rewrite_as_lerchphi9sc K sÃddlm}m}m}|j\}}|dkrF|d|ƒ S|jr¶|dkr¶|dƒ}|d|}x(t| ƒD]} ||j|ƒ}qƒW||ƒj||ƒSt ||ƒS(Niÿÿÿÿ(R t expand_mulRiitu( RR RKRRRRRRR"( R%R&R RKRR(R'RLR*t_((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyR#<s  (R=R>R?t classmethodRIR6RJR#(((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyR"Îs @#  RcB sMeZdZedd„ƒZdd„Zdd„Zd„Zdd„Z RS(sÊ Hurwitz zeta function (or Riemann zeta function). For `\operatorname{Re}(a) > 0` and `\operatorname{Re}(s) > 1`, this function is defined as .. math:: \zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s}, where the standard choice of argument for :math:`n + a` is used. For fixed :math:`a` with `\operatorname{Re}(a) > 0` the Hurwitz zeta function admits a meromorphic continuation to all of :math:`\mathbb{C}`, it is an unbranched function with a simple pole at :math:`s = 1`. Analytic continuation to other :math:`a` is possible under some circumstances, but this is not typically done. The Hurwitz zeta function is a special case of the Lerch transcendent: .. math:: \zeta(s, a) = \Phi(1, s, a). This formula defines an analytic continuation for all possible values of :math:`s` and :math:`a` (also `\operatorname{Re}(a) < 0`), see the documentation of :class:`lerchphi` for a description of the branching behavior. If no value is passed for :math:`a`, by this function assumes a default value of :math:`a = 1`, yielding the Riemann zeta function. See Also ======== dirichlet_eta, lerchphi, polylog References ========== .. [1] http://dlmf.nist.gov/25.11 .. [2] https://en.wikipedia.org/wiki/Hurwitz_zeta_function Examples ======== For :math:`a = 1` the Hurwitz zeta function reduces to the famous Riemann zeta function: .. math:: \zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}. >>> from sympy import zeta >>> from sympy.abc import s >>> zeta(s, 1) zeta(s) >>> zeta(s) zeta(s) The Riemann zeta function can also be expressed using the Dirichlet eta function: >>> from sympy import dirichlet_eta >>> zeta(s).rewrite(dirichlet_eta) dirichlet_eta(s)/(1 - 2**(1 - s)) The Riemann zeta function at positive even integer and negative odd integer values is related to the Bernoulli numbers: >>> zeta(2) pi**2/6 >>> zeta(4) pi**4/90 >>> zeta(-1) -1/12 The specific formulae are: .. math:: \zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!} .. math:: \zeta(-n) = -\frac{B_{n+1}}{n+1} At negative even integers the Riemann zeta function is zero: >>> zeta(-4) 0 No closed-form expressions are known at positive odd integers, but numerical evaluation is possible: >>> zeta(3).n() 1.20205690315959 The derivative of :math:`\zeta(s, a)` with respect to :math:`a` is easily computed: >>> from sympy.abc import a >>> zeta(s, a).diff(a) -s*zeta(s + 1, a) However the derivative with respect to :math:`s` has no useful closed form expression: >>> zeta(s, a).diff(s) Derivative(zeta(s, a), s) The Hurwitz zeta function can be expressed in terms of the Lerch transcendent, :class:`sympy.functions.special.lerchphi`: >>> from sympy import lerchphi >>> zeta(s, a).rewrite(lerchphi) lerchphi(1, s, a) cC sß|dkr0ttt|dfƒƒ\}}n!ttt||fƒƒ\}}|jr˜|tjkrptjS|tjkr˜|dk r˜||ƒSn|jr|tjkr·tjS|tjkrÍtjS|tj krçtj |S|tjkrtj Sn|j rÛ|j rÛ|jr@d|t| dƒ| d}n`|jrœ|jrœt|ƒt|ƒ}}d|ddd|d|t||}ndS|jrÀ|tt|ƒ|ƒS|t|d|ƒSqÛndS(Niiÿÿÿÿi(tNonetlisttmapRt is_NumberRtNaNRFtInfinityRCRDtComplexInfinityt is_integerRt is_negativeR tis_event is_positiveR RR tabs(RHR'ta_R)RtBtF((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyRIºs8 $!        %1 icK s7|dkr|S|jd}t|ƒddd|S(Niii(RRB(R%R(R)R:((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyt_eval_rewrite_as_dirichlet_etaás  cK std||ƒS(Ni(R(R%R(R)R:((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyRJçscC s)|jddj}|dk r%| SdS(Nii(Rtis_zeroRO(R%t arg_is_one((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyt_eval_is_finiteês cC sft|jƒdkr'|j\}}n|jd\}}|dkr\| t|d|ƒSt‚dS(Nii(i(tlenRRR(R%R5R(R)((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyR6ïs  N( R=R>R?RNRORIR^RJRaR6(((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyRNsj&   RBcB s&eZdZed„ƒZd„ZRS(s  Dirichlet eta function. For `\operatorname{Re}(s) > 0`, this function is defined as .. math:: \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}. It admits a unique analytic continuation to all of :math:`\mathbb{C}`. It is an entire, unbranched function. See Also ======== zeta References ========== .. [1] https://en.wikipedia.org/wiki/Dirichlet_eta_function Examples ======== The Dirichlet eta function is closely related to the Riemann zeta function: >>> from sympy import dirichlet_eta, zeta >>> from sympy.abc import s >>> dirichlet_eta(s).rewrite(zeta) (1 - 2**(1 - s))*zeta(s) cC sI|dkrtdƒSt|ƒ}|jtƒsEddd||SdS(Nii(R RR7(RHR(R'((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyRIs    cK sddd|t|ƒS(Nii(R(R%R(R:((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyR;#s(R=R>R?RNRIR;(((sElib/python2.7/site-packages/sympy/functions/special/zeta_functions.pyRBúst stieltjescB s eZdZedd„ƒZRS(syRepresents Stieltjes constants, :math:`\gamma_{k}` that occur in Laurent Series expansion of the Riemann zeta function. Examples ======== >>> from sympy import stieltjes >>> from sympy.abc import n, m >>> stieltjes(n) stieltjes(n) zero'th stieltjes constant >>> stieltjes(0) EulerGamma >>> stieltjes(0, 1) EulerGamma For generalized stieltjes constants >>> stieltjes(n, m) stieltjes(n, m) Constants are only defined for integers >= 0 >>> stieltjes(-1) zoo References ========== .. 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