g^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+GddeZ,GddeZ-GddeZ.GddeZ/GddeZ0GddeZ1edZ2dS)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)Polyc2eZdZdZdZd dZdZdZdZdS) 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 .. 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JKr/c 6|tSrY)r\r6rGr,r+r)kwargss r-_eval_rewrite_as_zetazlerchphi._eval_rewrite_as_zetas((...r/c 6|tSrY)r\rCr^s r-_eval_rewrite_as_polylogz!lerchphi._eval_rewrite_as_polylogs((111r/Nr1) __name__ __module__ __qualname____doc__rErWr\r`rbr%r/r-r r srggR?!?!?!B%%%%///22222r/r cTeZdZdZedZd dZdZdZdZ d fd Z xZ S) rCa 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 c|jru|tjurt|S|tjurt | S|tjur tjS|dkrt}||vr||S|jr tjS| tj}|rt|S|dur>|tjur|d|z z S|tjur |d|z dzz S|jr|d|z z S| ttr<|s!t|tjkdkr||t|SdSdS)Nr2Fr"T) is_numberr rAr6 NegativeOne dirichlet_etar; _dilogtableis_zeroequalsrZrrrr)clsr+r, dilogtablezones r-evalz polylog.eval$sf ; )AEzzAwwam##%a((((afv a(]]  ??%a=( 9 6Mxx  !77N U]] AF{{!a%y am##!a%!|#y !!a%y  55J ' ' )T )c!ffo$5N5N3q*Q--(( ( ) )5N5Nr/r"c\|j\}}|dkrt|dz ||z St)Nr2r")r9rCr)rGrVr+r,s r-rWz polylog.fdiffKs5y1 q==1q5!$$Q& &  r/c *|t||dzSNr"r )rGr+r,r_s r-_eval_rewrite_as_lerchphiz!polylog._eval_rewrite_as_lerchphiQs!Q""""r/c L|j\}}|dkrtd|z  S|jrk|dkretd}|d|z z }t | D]}|||z}t |||St||S)Nr"ru) r9rr:r rBr>rr?rC)rGrHr+r,ryrI_s r-rEzpolylog._eval_expand_funcTsy1 66AJJ;  < 0AFFc Aq1uIEA2YY ( (%**Q--e$$))!Q// /q!}}r/c2|jd}|jrdSdS)Nr"Tr9rm)rGr,s r- _eval_is_zerozpolylog._eval_is_zero`s& IaL 9 4  r/rcddlm}|j\}}||d}|tjur.||dt|jrdnd}|j r | |\} } n#ttf$r|cYSwxYw| j rt|| z } |||z|} |||||} | tjur| S| }|g}t%d| D]"}|| z}||||zz #t)|| zSt+t,|||||S)Nr)Order-+)dirr2)sympy.series.orderrr9r?r NaNlimitr is_negativermleadterm ValueErrorNotImplementedError is_positiver _eval_nseriesremoveOr;rBrFrsuperrC)rGxr*logxcdirrnur,z0rzrnewnortermr+r( __class__s r-rzpolylog._eval_nserieses,,,,,, A VVAq\\ ;;A"T((*>#G33CHHB : # A33 34      #qu~~E!Q$NNOOAq$55==??;;HFq$))AAIDHHT!R%Z((((Aw{"Wd##11!QdCCCs,BBBr1)r) rcrdrerf classmethodrrrWrwrEr}r __classcell__rs@r-rCrCsCCJ$)$)[$)L!!!! ###    DDDDDDDDDDr/rCcheZdZdZed dZddZddZddZdZ d Z dd Z dfd Z xZ S)r6a 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 ========== .. 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" !r/c .|jd}t|jdkr |jdn tj}|jrO|jr#t |t|dz |z S|jr|jdus |jdur tj S|S)Nrr"F) r9lenr rArrr6rrr)rGrHr+r)s r-rEzzeta._eval_expand_func#s IaL NNQ..DIaLLAE < } 2Aww!A#q!1!111 Q\U%:%:$--u  r/ct|jdkr |j\}}n |jdz\}}|dkr| t|dz|zSt)Nr2r1r")rr9r6r)rGrVr+r)s r-rWz zeta.fdiff.sZ ty>>Q  9DAqq9t#DAq q==2d1q5!nn$ $$ $r/rcPt|jdkr |j\}}n|jtjfz\}} ||\}}n#t $r|cYSwxYw|jr|jst tt| |||S)Nr2) rr9r rArrrrrr6_eval_as_leading_term) rGrrrr+r)rKers r-rzzeta._eval_as_leading_term8s ty>>Q  9DAqq9x'DAq ::a==DAqq"   KKK  = & &% %T4  66q$EEEsA A&%A&rYr1)Nr)rcrdrerfrrrrrrwrrErWrrrs@r-r6r6shhT[4)))) 1111 !!!!"""   %%%%FFFFFFFFFFr/r6cLeZdZdZeddZd dZejfdZ dZ dS) rka 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 Nc|tjur ||S|N|dkrtdSt|}|tsddd|z zz |zSdS|dkr3ddlm}td||z ||dzdz zSt||}t||dzdz }|ts(|ts|dd|z z|zz SdSdS)Nr"r2rdigamma)r rArr6rZ'sympy.functions.special.gamma_functionsr)ror+r)r,rz1z2s r-rrzdirichlet_eta.evalxs) ::3q66M 9Avv1vv QA55;; *A!H )) F !VV G G G G G Gq66GGAJJ&!A#q)9)99 9 !QZZ !ac1W  vvd|| &BFF4LL &AaC2 % % & & & &r/r"c ddlm}|dkrHttdt |dfddd|z zz t |zdfSttd||z ||dzdz zt |dft ||dd|z zt ||dzdz zz dfS)Nrrr"r2T)rrrrr r6rGr+r)r_rs r-r`z#dirichlet_eta._eval_rewrite_as_zetasCCCCCC 66c!ffbAhh/1q1Q3x<4772JD1QRR R#a&&771::-1a0@0@@"Q((Kaa!A#ha!A#q)9)9994@BB Br/c ddlm}ttd||z ||dzdz zt |dft d|z |d|dz zz dfS)Nrrr2r"T)rrrrr rrs r-_eval_rewrite_as_genocchiz'dirichlet_eta._eval_rewrite_as_genocchisCCCCCC#a&&771::-1a0@0@@"Q((K!A#q!!Q!A#Y/688 8r/ctd|jDr-|t|SdS)Nc3$K|] }|jV dSrY)ri)r'is r- z,dirichlet_eta._eval_evalf..s$..qq{......r/)allr9rewriter6 _eval_evalf)rGprecs r-rzdirichlet_eta._eval_evalfsM ..DI... . . 8<<%%11$77 7 8 8r/rYr1) rcrdrerfrrrr`r rArrr%r/r-rkrkIs,,\&&&[&$BBBB./U8888 88888r/rkc.eZdZdZedZdZdS) 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 cddlm}t|}|tjtjfvr tjSt|ts+||dz z||dz z|zdt|dz zzz SdSNr)gammar"r2) rrr6r r;rAHalfrDr)ror+rr,s r-rrzriemann_xi.evalsAAAAAA GG   6M!T"" 8a!e9UU1Q3ZZ')1R!A#Y;7 7 8 8r/c ~ddlm}||dz z||dz zt|zdt|dz zzz Sr)rrr6r)rGr+r_rs r-r`z riemann_xi._eval_rewrite_as_zetasOAAAAAA!a%yqs#DGG+QrAaCy[99r/N)rcrdrerfrrrr`r%r/r-rrsH.88[8:::::r/rc*eZdZdZeddZdS) 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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