o *ήcbS@sdZddlmZddlmZmZmZmZmZddl m Z m Z ddl m Z ddlmZmZmZddlmZmZmZmZddlmZmZmZdd lmZmZdd lmZdd l m!Z!Gd d d eZ"GdddeZ#GdddeZ$GdddeZ%GdddeZ&GdddeZ'dS)z$ Riemann zeta and related function. )Add)FunctionSsympifypiI)ArgumentIndexError expand_mul)Dummy) bernoulli factorialharmonic)re unpolarifyAbs polar_lift)log exp_polarexp)ceilingfloor)sqrt)Polyc@s:eZdZdZddZdddZddZd d Zd d Zd S)lerchphia] Lerch transcendent (Lerch phi function). Explanation =========== For $\operatorname{Re}(a) > 0$, $|z| < 1$ 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 $n + a$, 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 $\mathbb{C} - [1, \infty)$. Finally, for $x \in (1, \infty)$ 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 $\log{x}$ and $\log{\log{x}}$ (a branch of $\log{\log{x}}$ is needed to evaluate $\log{x}^{s-1}$). This concludes the analytic continuation. The Lerch transcendent is thus branched at $z \in \{0, 1, \infty\}$ and $a \in \mathbb{Z}_{\le 0}$. For fixed $z, a$ outside these branch points, it is an entire function of $s$. 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 $z=1$, 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 $z$ is a root of unity, the Lerch transcendent reduces to a sum of Hurwitz zeta functions: >>> expand_func(lerchphi(-1, s, a)) zeta(s, a/2)/2**s - zeta(s, a/2 + 1/2)/2**s If $a=1$, the Lerch transcendent reduces to the polylogarithm: >>> expand_func(lerchphi(z, s, 1)) polylog(s, z)/z More generally, if $a$ 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 $z$ and $a$ 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) 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 c  s|j\dkrtSjrIdkrItd}t| |dd|}tj}tD]}|||7}|| |}q3| |Sj rtj}tj }dkr|t krbd88 }tfddtD}n#dkrt d7}tfddtD}tjjg\tdtt d||dt fddtDSttrjdttj sd tt fvr;d krtddg\n2tkrtdd g\n$t krtd d g\njddtt}t|j|jg\tfd dtDStS) Nrtcs&g|]}| |qSr.0kanszrM/tmp/pip-target-vg8gfxp4/lib/python/sympy/functions/special/zeta_functions.py s&z.lerchphi._eval_expand_func..cs,g|]}d||dqSrrrr rr%r&s,cs>g|]}t|jdit|qS)r)polylog_eval_expand_funcrr)hintsmrootr#zetrr%r&s csBg|]}tdtt|t|qS)r()rrrzetar)r!pqr#rr%r&s:)argsr1 is_Integerr rrZeroreversed all_coeffsdiffsubs is_RationalOnerrranger2r3rrr isinstancerr) selfr+rstartrescaddmulargr) r!r+r,r"r2r3r-r#r$r.r%r*}s`      "  0   zlerchphi._eval_expand_funcrcCsZ|j\}}}|dkr| t||d|S|dkr+t||d||t||||St)Nr)r4rr)r?argindexr$r#r!rrr%fdiffs $zlerchphi.fdiffcCs|}||r |S|SN)r*has)r?r$r#r!targetrArrr%_eval_rewrite_helpers zlerchphi._eval_rewrite_helpercK||||tSrI)rLr1r?r$r#r!kwargsrrr%_eval_rewrite_as_zetazlerchphi._eval_rewrite_as_zetacKrMrI)rLr)rNrrr%_eval_rewrite_as_polylogrQz!lerchphi._eval_rewrite_as_polylogNr') __name__ __module__ __qualname____doc__r*rHrLrPrRrrrr%rsi <  rcsPeZdZdZeddZdddZddZd d Zd d Z dfdd Z Z S)r)a Polylogarithm function. Explanation =========== For $|z| < 1$ and $s \in \mathbb{C}$, 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 $n$. It admits an analytic continuation which is branched at $z=1$ (notably not on the sheet of initial definition), $z=0$ and $z=\infty$. The name polylogarithm comes from the fact that for $s=1$, 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). Examples ======== For $z \in \{0, 1, -1\}$, 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 $s$ is a negative integer, $0$ or $1$, 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 $z$ 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) See Also ======== zeta, lerchphi cCs |tjur t|S|tjurt| S|tjurtjS|dkr|tjkr2tddtdddS|dkrDtddt ttdS|t dd dkrdtd dtt dddddS|t dd dkrtd dtt ddddS|dt ddkrtddtt ddddS|t dddkrtddtt ddddS|j rtjS| tj}|rt|S|d ur|tjur|d|S|tjur|d|dS|j r|d|S| ttr |st|tjkd kr||t|SdSdS) Nr( r0r rFFT)rr<r1 NegativeOne dirichlet_etar6Halfrrrris_zeroequalsrJrrrr)clsr#r$zonerrr%evalsD     *&$$     (z polylog.evalrcCs(|j\}}|dkrt|d||St)Nr(r)r4r)r)r?rGr#r$rrr%rHIs z polylog.fdiffcKs|t||dSNrr)r?r#r$rOrrr%_eval_rewrite_as_lerchphiOrQz!polylog._eval_rewrite_as_lerchphicKsz|j\}}|dkrtd| S|jr8|dkr8td}|d|}t| D] }|||}q&t|||St||S)Nrru) r4rr5r r=r9r r:r))r?r+r#r$rfr@_rrr%r*Rs   zpolylog._eval_expand_funccCs|jd}|jr dSdS)NrTr4r^)r?r$rrr% _eval_is_zero^s zpolylog._eval_is_zerorc sddlm}|j\}}||d}|tjur%|j|dt|jr!dndd}|j rz | |\} } Wn t t fy>|YSw| j rt|| } ||||} |||||} | tjur`| S| }|g}td| D]}|| 9}||||qjt|| Stt|||||S)Nr)Order-+)dirr()sympy.series.orderrjr4r:rNaNlimitr is_negativer^leadterm ValueErrorNotImplementedError is_positiver _eval_nseriesremoveOr6r=appendrsuperr))r?xr"logxcdirrjnur$z0rgrnewnortermr#r __class__rr%rvcs0       zpolylog._eval_nseriesr')r) rSrTrUrV classmethodrbrHrer*rirv __classcell__rrrr%r)sE  . r)c@sDeZdZdZedddZdddZddd Zd d Zdd d Z dS)r1a Hurwitz zeta function (or Riemann zeta function). Explanation =========== 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 $n + a$ is used. For fixed $a$ with $\operatorname{Re}(a) > 0$ the Hurwitz zeta function admits a meromorphic continuation to all of $\mathbb{C}$, it is an unbranched function with a simple pole at $s = 1$. Analytic continuation to other $a$ 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 $s$ and $a$ (also $\operatorname{Re}(a) < 0$), see the documentation of :class:`lerchphi` for a description of the branching behavior. If no value is passed for $a$, by this function assumes a default value of $a = 1$, yielding the Riemann zeta function. Examples ======== For $a = 1$ 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 $\zeta(s, a)$ with respect to $a$ can be computed: >>> from sympy.abc import a >>> zeta(s, a).diff(a) -s*zeta(s + 1, a) However the derivative with respect to $s$ 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:`~.lerchphi`: >>> from sympy import lerchphi >>> zeta(s, a).rewrite(lerchphi) lerchphi(1, s, a) See Also ======== dirichlet_eta, lerchphi, polylog References ========== .. [1] http://dlmf.nist.gov/25.11 .. [2] https://en.wikipedia.org/wiki/Hurwitz_zeta_function NcCsr|durttt|df\}}n ttt||f\}}|jr3|tjur&tjS|tjur3|dur3||S|jrV|tjur>tjS|tjurFtjS|jrNtj |S|tjurVtj S|j r|j r|j rqtj|t| d| d}n)|jr|jrt|t|}}tj|ddd|d|t||}ndS|j r|tt||S|t|d|S|jrtj |SdSNrr()listmapr is_Numberrror<Infinityr^r]ComplexInfinity is_integerr5rqr[r is_evenrur rr abs)r`r$a_r!r1BFrrr%rbs>     $ 0 z zeta.evalrcKs.|dkr|S|jd}t|ddd|S)Nrrr()r4r\r?r#r!rOrrr%_eval_rewrite_as_dirichlet_etas z#zeta._eval_rewrite_as_dirichlet_etacKs td||Srcrdrrrr%re%s zzeta._eval_rewrite_as_lerchphicCs"|jddj}|dur| SdS)Nrrrh)r? arg_is_onerrr%_eval_is_finite(szzeta._eval_is_finitecCsHt|jdkr |j\}}n|jd\}}|dkr"| t|d|St)Nr(r'r)lenr4r1r)r?rGr#r!rrr%rH-s  z zeta.fdiffrIr') rSrTrUrVrrbrrerrHrrrr%r1sn  ' r1c@$eZdZdZeddZddZdS)r\a Dirichlet eta function. Explanation =========== 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 $\mathbb{C}$. It is an entire, unbranched 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) See Also ======== zeta References ========== .. [1] https://en.wikipedia.org/wiki/Dirichlet_eta_function cCs:|dkrtdSt|}|tsddd||SdSr)rr1rJ)r`r#r$rrr%rb\s  zdirichlet_eta.evalcKsddd|t|Sr)r1)r?r#rOrrr%rPdsz#dirichlet_eta._eval_rewrite_as_zetaNrSrTrUrVrrbrPrrrr%r\8s #  r\c@r) riemann_xia Riemann Xi function. Examples ======== The Riemann Xi function is closely related to the Riemann zeta function. The zeros of Riemann Xi function are precisely the non-trivial zeros of the zeta function. >>> from sympy import riemann_xi, zeta >>> from sympy.abc import s >>> riemann_xi(s).rewrite(zeta) s*(s - 1)*gamma(s/2)*zeta(s)/(2*pi**(s/2)) References ========== .. [1] https://en.wikipedia.org/wiki/Riemann_Xi_function cCsdddlm}t|}|tjtjfvrtjSt|ts0||d||d|dt|dSdSNr)gammarr() 'sympy.functions.special.gamma_functionsrr1rr6r<r]r>r)r`r#rr$rrr%rbs  ,zriemann_xi.evalcKs<ddlm}||d||dt|dt|dSr)rrr1r)r?r#rOrrrr%rPs 0z riemann_xi._eval_rewrite_as_zetaNrrrrr%rhs   rc@seZdZdZedddZdS) stieltjesa Represents Stieltjes constants, $\gamma_{k}$ 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) The 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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