o o h^@s\dZddlmZddlmZddlmZmZmZddl m Z m Z m Z ddl mZddlmZddlmZdd lmZdd lmZmZmZmZdd lmZmZmZmZdd lm Z m!Z!m"Z"dd l#m$Z$m%Z%ddl&m'Z'ddl(m)Z)ddl*m+Z+GdddeZ,GdddeZ-GdddeZ.GdddeZ/GdddeZ0GdddeZ1eddZ2dS) z$ Riemann zeta and related function. )Add)cacheit)ArgumentIndexError expand_mulFunction)piIInteger)Eq)S)Dummy)sympify) bernoulli factorialgenocchiharmonic)re unpolarifyAbs polar_lift)log exp_polarexp)ceilingfloor)sqrt) Piecewise)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] https://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} t| } g} tD]%} t| | | ttrۈjd i|| | | | |q||dt| Sttrjdttj sdtt fvradkrtddg\n2tkr-tdd g\n$t krs&z.lerchphi._eval_expand_func..cs,g|]}d||dqSrr!r"r%r!r*r+s,csBg|]}tdtt|t|qS)r-)rrrzetar")r&pqr(r!r*r+s:r!)argsr0 is_Integerr rr Zeroreversed all_coeffsdiffsubs is_RationalOnerrranger1r2rrrrpolylog isinstance_eval_expand_funcappendrr)selfhintsr startrescaddmulmzetrootup_zetaddargsr$argr!)r&r'r1r2r(r)r*r?sf      "    4    zlerchphi._eval_expand_funcrcCsZ|j\}}}|dkr| t||d|S|dkr+t||d||t||||St)Nr)r3rr)rAargindexr)r(r&r!r!r*fdiffs $zlerchphi.fdiffcCs|}||r |S|SN)r?has)rAtargetrDr!r!r*_eval_rewrite_helpers zlerchphi._eval_rewrite_helpercK |tSrQ)rTr0rAr)r(r&kwargsr!r!r*_eval_rewrite_as_zeta zlerchphi._eval_rewrite_as_zetacKrUrQ)rTr=rVr!r!r*_eval_rewrite_as_polylogrYz!lerchphi._eval_rewrite_as_polylogNr,) __name__ __module__ __qualname____doc__r?rPrTrXrZr!r!r!r*rsi A  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|jr-|tjur t|S|tjurt| S|tjurtjS|dkr-t}||vr-||S|jr3tjS| tj}|r?t|S|durd|tjurN|d|S|tjur[|d|dS|jrd|d|S| t t r||sut |tjkdkr~||t|SdSdS)Nr-FrT) is_numberr r;r0 NegativeOne dirichlet_etar5 _dilogtableis_zeroequalsrRrrrr)clsr(r) dilogtablezoner!r!r*eval$s4         "z polylog.evalrcCs(|j\}}|dkrt|d||St)Nr-r)r3r=r)rArOr(r)r!r!r*rPKs z polylog.fdiffcKs|t||dSNrr)rAr(r)rWr!r!r*_eval_rewrite_as_lerchphiQsz!polylog._eval_rewrite_as_lerchphicKsz|j\}}|dkrtd| S|jr8|dkr8td}|d|}t| D] }|||}q&t|||St||S)Nrru) r3rr4r r<r8rr9r=)rArBr(r)rlrC_r!r!r*r?Ts   zpolylog._eval_expand_funccCs|jd}|jr dSdS)NrTr3rc)rAr)r!r!r* _eval_is_zero`s zpolylog._eval_is_zerorc sddlm}|j\}}||d}|tjur%|j|dt|jr!dndd}|j rz | |\} } Wn t t fy>|YSw| j rt|| } ||||} |||||} | tjur`| S| }|g}td| D]}|| 9}||||qjt|| Stt|||||S)Nr)Order-+)dirr-)sympy.series.orderrpr3r9r NaNlimitr is_negativercleadterm ValueErrorNotImplementedError is_positiver _eval_nseriesremoveOr5r<r@rsuperr=)rAxr'logxcdirrpnur)z0rmrnewnortermr(r$ __class__r!r*r|es0       zpolylog._eval_nseriesr,)r) r[r\r]r^ classmethodrhrPrkr?ror| __classcell__r!r!rr*r=sE  & r=csheZdZdZedddZdddZddd Zdd d Zd d Z ddZ dddZ dfdd Z Z S)r0a 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 $a$ not a nonpositive integer the Hurwitz zeta function admits a meromorphic continuation to all of $\mathbb{C}$; it is an unbranched function with a simple pole at $s = 1$. 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 $a$ a default value of $a = 1$ is assumed, 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 nonnegative even and negative integer values is related to the Bernoulli numbers and polynomials: >>> zeta(2) pi**2/6 >>> zeta(4) pi**4/90 >>> zeta(0) -1/2 >>> zeta(-1) -1/12 >>> zeta(-4) 0 The specific formulae are: .. math:: \zeta(2n) = -\frac{(2\pi i)^{2n} B_{2n}}{2(2n)!} .. math:: \zeta(-n,a) = -\frac{B_{n+1}(a)}{n+1} 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] https://dlmf.nist.gov/25.11 .. [2] https://en.wikipedia.org/wiki/Hurwitz_zeta_function NcCs*|tjur ||S|tjus|tjurtjS|tjurtjS|tjur&tjS|tjur.tjS|j}|dur8tj}|rH|jrHtd|||dS|tjuri|re|j rgdt t | t|dt |SdSdS|r||jr||j r|||t|d|S|jr|jr|jdus|jdurtjSdSdSdS)Nrr-F)r r;ruComplexInfinityInfinityr5r4is_nonpositiveris_evenrrrr{r is_integer)rer(r&sintr!r!r*rhs4       & z zeta.evalrcKsV|dkr |jr |jr |jr dtt| t|dt|Std|||dS)Nrr-)ris_nonnegativerrrrrrAr(r&rWr!r!r*_eval_rewrite_as_bernoullis&zzeta._eval_rewrite_as_bernoullicKs.|dkr|S|jd}t|ddd|S)Nrrr-)r3rarr!r!r*_eval_rewrite_as_dirichlet_etas z#zeta._eval_rewrite_as_dirichlet_etacKs td||Srirjrr!r!r*rks zzeta._eval_rewrite_as_lerchphicCs"|jddj}|dur| SdS)Nrrrn)rA arg_is_oner!r!r*_eval_is_finiteszzeta._eval_is_finitecKsn|jd}t|jdkr|jdntj}|jr5|jr%t|t|d|S|jr5|jdus2|jdur5tj S|S)NrrF) r3lenr r;rr{r0rrru)rArBr(r&r!r!r*r?#s  zzeta._eval_expand_funccCsHt|jdkr |j\}}n|jd\}}|dkr"| t|d|St)Nr-r,r)rr3r0r)rArOr(r&r!r!r*rP.s  z zeta.fdiffrcszt|jdkr |j\}}n |jtjf\}}z ||\}}Wn ty*|YSw|jr3|js3ttt | |||S)Nr-) rr3r r;rxrzrwr{r~r0_eval_as_leading_term)rArrrr(r&rEerr!r*r8s   zzeta._eval_as_leading_termrQr,)Nr)r[r\r]r^rrhrrrkrr?rPrrr!r!rr*r0sj      r0c@s>eZdZdZed ddZd ddZejfdd Z d d Z dS)raa Dirichlet eta function. Explanation =========== For $\operatorname{Re}(s) > 0$ and $0 < x \le 1$, this function is defined as .. math:: \eta(s, a) = \sum_{n=0}^\infty \frac{(-1)^n}{(n+a)^s}. It admits a unique analytic continuation to all of $\mathbb{C}$ for any fixed $a$ not a nonpositive integer. It is an entire, unbranched function. It can be expressed using the Hurwitz zeta function as .. math:: \eta(s, a) = \zeta(s,a) - 2^{1-s} \zeta\left(s, \frac{a+1}{2}\right) and using the generalized Genocchi function as .. math:: \eta(s, a) = \frac{G(1-s, a)}{2(s-1)}. In both cases the limiting value of $\log2 - \psi(a) + \psi\left(\frac{a+1}{2}\right)$ is used when $s = 1$. Examples ======== >>> from sympy import dirichlet_eta, zeta >>> from sympy.abc import s >>> dirichlet_eta(s).rewrite(zeta) Piecewise((log(2), Eq(s, 1)), ((1 - 2**(1 - s))*zeta(s), True)) See Also ======== zeta References ========== .. [1] https://en.wikipedia.org/wiki/Dirichlet_eta_function .. [2] Peter Luschny, "An introduction to the Bernoulli function", https://arxiv.org/abs/2009.06743 NcCs|tjur ||S|dur*|dkrtdSt|}|ts(ddd||SdS|dkrDddlm}td||||ddSt||}t||dd}|tsf|tsh|dd||SdSdS)Nrr-rdigamma)r r;rr0rR'sympy.functions.special.gamma_functionsr)rer(r&r)rz1z2r!r!r*rhxs"     zdirichlet_eta.evalrcKsddlm}|dkr"ttdt|dfddd|t|dfSttd||||ddt|dft||dd|t||dddfS)Nrrrr-T)rrrrr r0rAr(r&rWrr!r!r*rXs 0**z#dirichlet_eta._eval_rewrite_as_zetacKsVddlm}ttd||||ddt|dftd||d|ddfS)Nrrr-rT)rrrrr rrr!r!r*_eval_rewrite_as_genocchis *z'dirichlet_eta._eval_rewrite_as_genocchicCs(tdd|jDr|t|SdS)Ncss|]}|jVqdSrQ)r_)r#ir!r!r* sz,dirichlet_eta._eval_evalf..)allr3rewriter0 _eval_evalf)rAprecr!r!r*rszdirichlet_eta._eval_evalfrQr,) r[r\r]r^rrhrXr r;rrr!r!r!r*raIs.   rac@s$eZdZdZeddZddZdS) 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-) rrr0r r5r;Halfr>r)rer(rr)r!r!r*rhs  ,zriemann_xi.evalcKs<ddlm}||d||dt|dt|dSr)rrr0r)rAr(rWrr!r!r*rXs 0z riemann_xi._eval_rewrite_as_zetaN)r[r\r]r^rrhrXr!r!r!r*rs   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 NcCs|durt|}|tjurtjS|jr|jrtjS|jr=|tjur$tjS|dkr+tjS|js1tjS|tjur=|dvr=tjS|j rCtjS|j rM|dvrMtjS|j dkrUtjSdS)NrriF) r r rur4rr is_Numberr5 EulerGammais_extended_negativercr)rer'r&r!r!r*rhs,    zstieltjes.evalrQ)r[r\r]r^rrhr!r!r!r*rs$rcCstjtddtdddtdtddtttdtdd dtd dttdddddtdd dtd dttdddddtddtddttddddtdddtddttddddtttjtdd t t tjtdd dttdd ttjttdtddttdd ttjttdtddtdtdd d tttdd dtdd ttji S) Nr- r/r rN0`)r rrrr rrCatalanr!r!r!r*rbs 8400..HrbN)3r^sympy.core.addrsympy.core.cachersympy.core.functionrrrsympy.core.numbersrrr sympy.core.relationalr sympy.core.singletonr sympy.core.symbolr sympy.core.sympifyr %sympy.functions.combinatorial.numbersrrrr$sympy.functions.elementary.complexesrrrr&sympy.functions.elementary.exponentialrrr#sympy.functions.elementary.integersrr(sympy.functions.elementary.miscellaneousr$sympy.functions.elementary.piecewisersympy.polys.polytoolsrrr=r0rarrrbr!r!r!r*s6         G-@S'B