o 8VaS@sdZddlmZddlmZmZmZmZmZddl m Z ddl m Z m Z mZddlmZddlmZmZddlmZGd d d eZGd d d eZGd ddeZGdddeZGdddeZGdddeZdS)z$ Riemann zeta and related function. )Add)FunctionSsympifypiI)ArgumentIndexError) bernoulli factorialharmonic)re)log exp_polar)sqrtc@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 sddlmmm}m}m}m}m}m |j \  dkr#t S j r] dkr]|d}|| |dd|}t j } tD]} | | |7} |||}qG| | Sjrt j } t j} dkr|krvd88  } | fddtD} n#dkr| d7 } | fddtD} t jjg\|dt d| |  d| fd dtDSt r j dtjs d fvrT d krt ddg\n2 krt dd g\n$ kr-t d d g\n j ddt} t | j| jg\| fd dtDSt S) Nr)exprfloorrPolyDummyr unpolarifytcs&g|]}| |qSr.0kanszrH/usr/lib/python3/dist-packages/sympy/functions/special/zeta_functions.py s&z.lerchphi._eval_expand_func..cs,g|]}d||dqSrrrrrr!r"s,cs>g|]}t|jdi|qS)r)polylog_eval_expand_funcr)hintsmrootrrzetrr!r"s csBg|]}dt|t|qS)r$)rzetar)rrrpqrrr!r"s:)sympyrrrrrrrrargsr- is_IntegerrZeroreversedZ all_coeffsdiffsubsZ is_RationalOneranger.r/r isinstancer)selfr'rrrrrrstartrescaddmulargr) rrrr'r(rr.r/r)rrr r*r!r&zsb(      "  2    zlerchphi._eval_expand_funcrcCsZ|j\}}}|dkr| t||d|S|dkr+t||d||t||||St)Nr)r1rr)r:argindexr rrrrr!fdiffs $zlerchphi.fdiffcCs|}||r |S|SN)r&has)r:r rrtargetr<rrr!_eval_rewrite_helpers zlerchphi._eval_rewrite_helpercK||||tSrD)rGr-r:r rrkwargsrrr!_eval_rewrite_as_zetazlerchphi._eval_rewrite_as_zetacKrHrD)rGr%rIrrr!_eval_rewrite_as_polylogrLz!lerchphi._eval_rewrite_as_polylogNr#) __name__ __module__ __qualname____doc__r&rCrGrKrMrrrr!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 cCsDt||f\}}|tjurt|S|tjurt| S|tjur#tjS|dkr|tjkr:tddt dddS|dkrLtddt tt dS|t dd dkrltd dt t dddddS|t dd dkrtd dt t ddddS|dt ddkrtddt t ddddS|t dddkrtddt t ddddS|j rtjS| tj}|rt|S|d ur|tjur|d|S|tjur|d|dS|j r|d|Sd d lm}m}m}|t|r|s||tjkd kr ||||SdSdS) Nr$ r,r rAFr)Absr polar_liftT)rrr7r-Z NegativeOne dirichlet_etar3Halfrr rris_zeroZequals$sympy.functions.elementary.complexesrVrrWrEr)clsrr ZzonerVrrWrrr!evalsH     *&$$     (z polylog.evalrcCs(|j\}}|dkrt|d||St)Nr$r)r1r%r)r:rBrr rrr!rCJs z polylog.fdiffcKs|t||dSNrr)r:rr rJrrr!_eval_rewrite_as_lerchphiPrLz!polylog._eval_rewrite_as_lerchphic Ksddlm}m}m}|j\}}|dkr|d| S|jrB|dkrB|d}|d|}t| D] } |||}q0||||St ||S)Nr)r expand_mulrru) r0r rarr1r2r8r5r6r%) r:r'r rarrr rbr;_rrr!r&Ss   zpolylog._eval_expand_funccCs|jd}|jr dSdS)NrTr1rZ)r:r rrr! _eval_is_zero`s zpolylog._eval_is_zerorc sddlm}m}|j\}}||d} | tjur'|j|dt|j r#dndd} | j rz | |\} } Wn t t fy@|YSw| jr||| } ||||} |||||}|tjurb| S|}|g}td| D]}||9}||||qlt|| Stt|||||S)Nr)ceilingOrder-+)dirr$)r0rfrgr1r6rNaNlimitr is_negativerZZleadterm ValueErrorNotImplementedError is_positive _eval_nseriesZremoveOr3r8appendrsuperr%)r:xrZlogxcdirrfrgZnur Zz0rcrZnewnorZtermrr __class__rr!rqes0      zpolylog._eval_nseriesr#)r) rNrOrPrQ classmethodr]rCr`r&rerq __classcell__rrrxr!r%sE  1 r%c@sDeZdZdZedddZdddZddd Zd d Zdd d Z dS)r-a 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 NcCsn|durttt|df\}}n ttt||f\}}|jr3|tjur&tjS|tjur3|dur3||S|jrV|tjur>tjS|tjurFtjS|jrNtj |S|tjurVtj S|j r|j r|j rpd|t| d| d}n(|jr|jrt|t|}}d|ddd|d|t||}ndS|j r|tt||S|t|d|S|jrtj |SdS)Nrr+r$)listmapr is_Numberrrkr7ZInfinityrZrYComplexInfinity is_integerr2rmr Zis_evenrpr rr abs)r\r Za_rr-BFrrr!r]s>     " . z zeta.evalrcKs.|dkr|S|jd}t|ddd|S)Nrrr$)r1rXr:rrrJrrr!_eval_rewrite_as_dirichlet_eta!s z#zeta._eval_rewrite_as_dirichlet_etacKs td||Sr^r_rrrr!r`'s zzeta._eval_rewrite_as_lerchphicCs"|jddj}|dur| SdS)Nrrrd)r:Z arg_is_onerrr!_eval_is_finite*szzeta._eval_is_finitecCsHt|jdkr |j\}}n|jd\}}|dkr"| t|d|St)Nr$r#r)lenr1r-r)r:rBrrrrr!rC/s  z zeta.fdiffrDr#) rNrOrPrQrzr]rr`rrCrrrr!r-sn  ' r-c@$eZdZdZeddZddZdS)rXa 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||SdSNrr$)r r-rE)r\rr rrr!r]^s  zdirichlet_eta.evalcKsddd|t|Sr)r-)r:rrJrrr!rKfsz#dirichlet_eta._eval_rewrite_as_zetaNrNrOrPrQrzr]rKrrrr!rX:s #  rXc@r) riemann_xia Riemann Xi function. Examples ======== The Riemann Xi function is closely related to the Riemann zeta function. The zeros of Riemann Xi function are precisely the non-trivial zeros of the zeta function. >>> from sympy import riemann_xi, zeta >>> from sympy.abc import s >>> riemann_xi(s).rewrite(zeta) s*(s - 1)*gamma(s/2)*zeta(s)/(2*pi**(s/2)) References ========== .. [1] https://en.wikipedia.org/wiki/Riemann_Xi_function cCshddlm}t|}|tjus|tjurtjSt|ts2||d||d|dt|dSdSNr)gammarr$) r0rr-rr3r7rYr9r)r\rrr rrr!r]s  ,zriemann_xi.evalcKs<ddlm}||d||dt|dt|dSr)r0rr-r)r:rrJrrrr!rKs 0z riemann_xi._eval_rewrite_as_zetaNrrrrr!rjs   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 ========== .. [1] https://en.wikipedia.org/wiki/Stieltjes_constants NcCst|}|durt|}|tjurtjS|jr|jrtjS|jrA|tjur(tjS|dkr/tjS|js5tjS|tjurA|dvrAtjS|j rGtjS|j rQ|dvrQtjS|j dkrYtjSdS)Nrr^F) rrrkr2Zis_nonpositiverr~r3Z EulerGammaZis_extended_negativerZr)r\rrrrr!r]s.    zstieltjes.evalrD)rNrOrPrQrzr]rrrr!rs$rN)rQr0rZ sympy.corerrrrrZsympy.core.functionrZ%sympy.functions.combinatorial.numbersr r r r[r Z&sympy.functions.elementary.exponentialr rZ(sympy.functions.elementary.miscellaneousrrr%r-rXrrrrrr!s"    C910&