o *ήci@sddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZmZmZmZmZmZmZmZddlmZmZmZdd lmZdd lmZmZmZm Z m!Z!dd l"m#Z#dd l$m%Z%dd l&m'Z'ddl(m)Z)m*Z*ddl+m,Z,ddl-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4ddl5m6Z6m7Z7ddl8m9Z9ddl:m;Z;GdddeZGdddedZ?ddZ@Gd d!d!eZAGd"d#d#eZBd$S)%)product)Tuple)Expr)sympify)Add)cacheit)FunctionArgumentIndexError expand_log expand_mul FunctionClass PoleErrorexpand_multinomialexpand_complex) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRationalpiI ImaginaryUnit)global_parameters)Pow)S)WildDummy) factorial)arg unpolarifyimreAbs)sqrt) multiplicity perfect_power) factorint)cancelc@seZdZdZejfZeddZdddZ ddZ ed d Z d d Z d dZ ddZddZddZddZddZddZddZdS)ExpBaseTcCs|jjSN)expkindselfr/M/tmp/pip-target-vg8gfxp4/lib/python/sympy/functions/elementary/exponential.pyr,)sz ExpBase.kindcCtS)z= Returns the inverse function of ``exp(x)``. logr.argindexr/r/r0inverse-zExpBase.inversecCs@|j}|j}|s| js|}|rtj|| fS|tjfS)a- Returns this with a positive exponent as a 2-tuple (a fraction). Examples ======== >>> from sympy import exp >>> from sympy.abc import x >>> exp(-x).as_numer_denom() (1, exp(x)) >>> exp(x).as_numer_denom() (exp(x), 1) )r+ is_negativecould_extract_minus_signrOnefunc)r.r+neg_expr/r/r0as_numer_denom3s  zExpBase.as_numer_denomcCs |jdS)z7 Returns the exponent of the function. r)argsr-r/r/r0r+Ks z ExpBase.expcCs|dt|jfS)z7 Returns the 2-tuple (base, exponent). r1)r<rr?r-r/r/r0 as_base_expRszExpBase.as_base_expcC||jSr*)r<r+adjointr-r/r/r0 _eval_adjointXzExpBase._eval_adjointcCrAr*)r<r+ conjugater-r/r/r0_eval_conjugate[rDzExpBase._eval_conjugatecCrAr*)r<r+ transposer-r/r/r0_eval_transpose^rDzExpBase._eval_transposecCs.|j}|jr|jr dS|jrdS|jrdSdSNTF)r+ is_infiniteis_extended_negativeis_extended_positive is_finiter.rr/r/r0_eval_is_finiteaszExpBase._eval_is_finitecCsJ|j|j}|j|jkr"|jj}|rdS|jjrt|r dSdSdS|jSrI)r<r?r+is_zero is_rationalr)r.szr/r/r0_eval_is_rationalks  zExpBase._eval_is_rationalcCs |jtjuSr*)r+rNegativeInfinityr-r/r/r0 _eval_is_zerov zExpBase._eval_is_zerocCs"|\}}tt||dd|S)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)r@r _eval_power)r.otherber/r/r0rZys zExpBase._eval_powerc s|ddlm}ddlm}jd}|jr$|jr$tfdd|jDSt ||r9|jr9| |j g|j RS |S)Nr)Product)Sumc3s|]}|VqdSr*)r<).0xr-r/r0 sz1ExpBase._eval_expand_power_exp..) sympy.concrete.productsr^sympy.concrete.summationsr_r?is_Addis_commutativerfromiter isinstancer<functionlimits)r.hintsr^r_rr/r-r0_eval_expand_power_exps     zExpBase._eval_expand_power_expNr1)__name__ __module__ __qualname__ unbranchedrComplexInfinity_singularitiespropertyr,r7r>r+r@rCrFrHrOrTrVrZrlr/r/r/r0r)$s$      r)c@s@eZdZdZdZdZddZddZdd Zd d Z d d Z dS) exp_polara< Represent a *polar number* (see g-function Sphinx documentation). Explanation =========== ``exp_polar`` represents the function `Exp: \mathbb{C} \rightarrow \mathcal{S}`, sending the complex number `z = a + bi` to the polar number `r = exp(a), \theta = b`. It is one of the main functions to construct polar numbers. Examples ======== >>> from sympy import exp_polar, pi, I, exp The main difference is that polar numbers do not "wrap around" at `2 \pi`: >>> exp(2*pi*I) 1 >>> exp_polar(2*pi*I) exp_polar(2*I*pi) apart from that they behave mostly like classical complex numbers: >>> exp_polar(2)*exp_polar(3) exp_polar(5) See Also ======== sympy.simplify.powsimp.powsimp polar_lift periodic_argument principal_branch TFcCstt|jdSNr)r+r"r?r-r/r/r0 _eval_Absszexp_polar._eval_AbscCsxt|jd}z |t kp|tk}Wn tyd}Ynw|r"|St|jd|}|dkr:t|dkr:t|S|S)z. Careful! any evalf of polar numbers is flaky rT)r!r?r TypeErrorr+ _eval_evalfr")r.precibadresr/r/r0rys zexp_polar._eval_evalfcCs||jd|Srv)r<r?)r.r[r/r/r0rZszexp_polar._eval_powercCs|jdjrdSdS)NrT)r?is_extended_realr-r/r/r0_eval_is_extended_reals z exp_polar._eval_is_extended_realcCs"|jddkr |tjfSt|Srv)r?rr;r)r@r-r/r/r0r@s  zexp_polar.as_base_expN) rnrorp__doc__is_polar is_comparablerwryrZrr@r/r/r/r0rus% ruc@seZdZddZdS)ExpMetacCs&t|jjvrdSt|to|jtjuS)NT)r+ __class____mro__rhrbaserExp1)clsinstancer/r/r0__instancecheck__s zExpMeta.__instancecheck__N)rnrorprr/r/r/r0rs rc@seZdZdZd,ddZddZeddZed d Z e e d d Z d-ddZ ddZddZddZddZddZd.ddZddZd/d d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+ZdS)0r+a9 The exponential function, :math:`e^x`. Examples ======== >>> from sympy import exp, I, pi >>> from sympy.abc import x >>> exp(x) exp(x) >>> exp(x).diff(x) exp(x) >>> exp(I*pi) -1 Parameters ========== arg : Expr See Also ======== log r1cCs|dkr|St||)z@ Returns the first derivative of this function. r1)r r5r/r/r0fdiffs z exp.fdiffcCsddlm}m}|jd}|jrctjtj}||| fvr tjS| tj tj}|re|| d|rg|| |r>tj S|||rHtjS|| |tjrVtj S|||tjritjSdSdSdSdS)Nr)askQ)sympy.assumptionsrrr?is_MulrrInfinityNaNas_coefficientPiintegerevenr;odd NegativeOneHalf)r. assumptionsrrrIoocoeffr/r/r0 _eval_refines*  zexp._eval_refinecCspddlm}ddlm}ddlm}ddlm}t||r!| St j r*t t j|S|jrU|t jur5t jS|jr;t jS|t jurCt jS|t jurKt jS|t jurSt jSnZ|t jur]t jSt|trg|jdSt||rw|t |jt |jSt||r||S|jrU|t jt j}|rd|j r|j!rt jS|j"rt j#S|t j$j!rt j S|t j$j"rt jSn|j%r|d}|dkr|d8}||kr||t jt jS|&\}}|t jt jfvr|j'r|t jur| }t(|jr|t jurt jSt(|j)r t*|t jur t jSt(|j+rt jSdS|gd} } t,-|D](} || } t| tr<| dur9| jd} q!dS| j.rG| /| q!dS| rS| t,| SdS|j0rg} g}d}|jD]:}|t jurq|/|qb||}t||r|jd|kr|/|jdd }qb|/|qb| /|qb| s|rt,| |t1|dd S|jrt jSdS) Nr AccumBounds) MatrixBaseSetExpr logcombinerr1FTrX)2sympy.calculusrsympy.matrices.matricesrsympy.sets.setexprrsympy.simplify.simplifyrrhr+r exp_is_powrrr is_NumberrrPr;rrUZerorrr4r?minmax _eval_funcrrrr is_integeris_evenis_oddrr is_Rational as_coeff_Mul is_numberr" is_positiver!r9r make_argsrappendrer)rrrrrrrncoefftermscoeffslog_termtermterm_outadd argchangedanewar/r/r0evals                              zexp.evalcCstjS)z? Returns the base of the exponential function. )rrr-r/r/r0r|szexp.basecGsT|dkrtjS|dkrtjSt|}|r"|d}|dur"|||S||t|S)zJ Calculates the next term in the Taylor series expansion. rN)rrr;rr)nraprevious_termspr/r/r0 taylor_terms zexp.taylor_termTcKstddlm}m}|jd\}}|r%|j|fi|}|j|fi|}||||}}t||t||fS)aJ Returns this function as a 2-tuple representing a complex number. Examples ======== >>> from sympy import exp, I >>> from sympy.abc import x >>> exp(x).as_real_imag() (exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x))) >>> exp(1).as_real_imag() (E, 0) >>> exp(I).as_real_imag() (cos(1), sin(1)) >>> exp(1+I).as_real_imag() (E*cos(1), E*sin(1)) See Also ======== sympy.functions.elementary.complexes.re sympy.functions.elementary.complexes.im r)cossin)(sympy.functions.elementary.trigonometricrrr? as_real_imagexpandr+)r.deeprkrrr"r!r/r/r0rszexp.as_real_imagcCs|jrt|jt|j}n |tjur|jrt}t|ts"|tjur1dd}t |||||S|turA|jsA||j ||St |||S)NcSs&|jst|trt|ddiS|S)NrYF)is_Powrhr+rr@)rr/r/r0s z exp._eval_subs..) rr+r4rrr is_Functionrhr _eval_subs_subsr)r.oldnewfr/r/r0rszexp._eval_subscCsF|jdjrdS|jdjr!td tj|jdtj}|jSdS)NrTr)r?r~ is_imaginaryrrrrr.arg2r/r/r0rs   zexp._eval_is_extended_realcCsdd}t||jdS)Ncss|jV|jVdSr*) is_complexrKrr/r/r0complex_extended_negatives z7exp._eval_is_complex..complex_extended_negativer)rr?)r.rr/r/r0_eval_is_complexszexp._eval_is_complexcCsJ|jtjtjjr dSt|jjr!|jjrdS|jtjjr#dSdSdSrI)r+rrrrQrrP is_algebraicr-r/r/r0_eval_is_algebraics zexp._eval_is_algebraiccCsB|jjr |jdtjuS|jjrtj |jdtj}|jSdSrv) r+r~r?rrUrrrrrr/r/r0_eval_is_extended_positives zexp._eval_is_extended_positiverc sddlmddlm}ddlm}ddlm}ddlm }|j } | j |||d} | j r0d| S|| |d} | tjurD||||S| tjurK|Stfd d | jDrY|Std } |} z|| j||d |}Wn ttfyzd}Ynw|r|dkr|||} t | | | }t | || | | }||t||kr|S|r|dkr||| | ||||d|7}n ||| | ||7}|}||d dd}dd}td|gd}|tj|ttj|}|S)Nrsignceiling)limitOrderpowsimprlogxr1c3s|] }t|tfVqdSr*)rhr)r`rrr/r0rbsz$exp._eval_nseries..trTr+rcombinecSs|jo|jdvS)N))rq)rar/r/r0rsz#exp._eval_nseries..w) properties) $sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimprr+ _eval_nseriesis_OrderremoveOrrUranyr?ras_leading_termgetnNotImplementedErrorr _taylorsubsr4rrreplacerr)r.rarrcdirrrrrr arg_seriesarg0rntermscf exp_seriesr simpleratrr/rr0rsL          (zexp._eval_nseriescCsNg}d}t|D]}|||jd|}|j||d}||qt|S)Nrr)rangerr?nseriesrrr)r.rarlgr{r/r/r0r s z exp._taylorNcCsddlm}|jdj||d}||d}|tjur tjSt||r5t |tj kr1t | St |S|tjur@| |d}|j durIt |Std|)NrrrFCannot expand %s around 0)sympy.calculus.utilrr?r(rrrrrhr"rr+rrJr )r.rarrrrrr/r/r0_eval_as_leading_terms         zexp._eval_as_leading_termcKs8ddlm}tj}|||tjd||||S)Nr)rr)rrrrr)r.rkwargsrrr/r/r0_eval_rewrite_as_sin* &zexp._eval_rewrite_as_sincKs8ddlm}tj}|||||||tjdS)Nr)rr)rrrrr)r.rrrrr/r/r0_eval_rewrite_as_cos/rzexp._eval_rewrite_as_coscKs,ddlm}d||dd||dS)Nr)tanhr1r)%sympy.functions.elementary.hyperbolicr)r.rrrr/r/r0_eval_rewrite_as_tanh4s  zexp._eval_rewrite_as_tanhcKsddlm}m}|jr9|tjtj}|r;|jr=|tj||tj|}}t ||s?t ||sA|tj|SdSdSdSdSdS)Nr)rr) rrrrrrrrrrh)r.rrrrrcosinesiner/r/r0_eval_rewrite_as_sqrt8s zexp._eval_rewrite_as_sqrtcKs@|jrdd|jD}|rt|djd||dSdSdS)NcSs(g|]}t|trt|jdkr|qSrm)rhr4lenr?)r`rr/r/r0 Cs(z,exp._eval_rewrite_as_Pow..r)rr?rr)r.rrlogsr/r/r0_eval_rewrite_as_PowAs zexp._eval_rewrite_as_PowrmTrrv)rnrorprrr classmethodrrtr staticmethodrrrrrrrrrrrrrrrr"r/r/r/r0r+s2   i     ,  r+) metaclasscCsR|jtjdd\}}|dkr|jr||fS|tj}|r'|jr'|jr'||fSdS)a Try to match expr with $a + Ib$ for real $a$ and $b$. ``match_real_imag`` returns a tuple containing the real and imaginary parts of expr or ``(None, None)`` if direct matching is not possible. Contrary to :func:`~.re()`, :func:`~.im()``, and ``as_real_imag()``, this helper will not force things by returning expressions themselves containing ``re()`` or ``im()`` and it does not expand its argument either. Tas_Addr)NN)as_independentrris_realr)exprr_i_r/r/r0match_real_imagHs  r/c@seZdZUdZeeed<ejej fZ d+ddZ d+ddZ e d,d d Zd d Zeed dZd-ddZddZd-ddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd.d'd(Zd/d)d*ZdS)0r4a The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp r?r1cCs |dkr d|jdSt||)z? Returns the first derivative of the function. r1r)r?r r5r/r/r0rs z log.fdiffcCr2)zC Returns `e^x`, the inverse function of `\log(x)`. )r+r5r/r/r0r7r8z log.inverseNcCsJddlm}ddlm}t|}|dur`t|}|dkr&|dkr#tjStjSzt||}|r=|t |||t |WSt |t |WSt yNYnw|tj ur\||||S||S|j r|j ritjS|tjurqtjS|tjurytjS|tjurtjS|tjurtjS|jr|jdkr||j S|jr|jtj ur|jjr|jStj}t|tr|jjr|jSt|tr|jjrt|j\}}|r|jr|dtj;}|tjkr|dtj8}|t||ddSn.r/)%sympy.concreter_r^getrr?r r< is_Integerr&r'keyssumitemsrr4rrrrrrhr_eval_expand_logr9rrrrrr+r~ris_nonpositiver rirj)r.rrkr_r^r=r>rrlogargrr,nonposrarr\r]r/r/r0rF>sp                  zlog._eval_expand_logcKsddlm}m}m}t|jdkr||j|jfi|S|||jdfi|}|dr3||}||dd}t||g|ddS) Nr)r simplifyinversecombinerr7Tr0measure)key)rr rJrKrr?r<r)r.rr rJrKr,r/r/r0_eval_simplifyws zlog._eval_simplifycKs|jd}|r|jdj|fi|}t|}||kr |tjfSt|}|ddr;d|d<t|j|fi||fSt||fS)a Returns this function as a complex coordinate. Examples ======== >>> from sympy import I, log >>> from sympy.abc import x >>> log(x).as_real_imag() (log(Abs(x)), arg(x)) >>> log(I).as_real_imag() (0, pi/2) >>> log(1 + I).as_real_imag() (log(sqrt(2)), pi/4) >>> log(I*x).as_real_imag() (log(Abs(x)), arg(I*x)) rr4Fcomplex)r?rr#rrrrAr4)r.rrksargsarg_abssarg_argr/r/r0rs    zlog.as_real_imagcCs^|j|j}|j|jkr,|jddjrdS|jdjr(t|jddjr*dSdSdS|jSNrr1TF)r<r?rPrQrr.rRr/r/r0rTs   zlog._eval_is_rationalcCs^|j|j}|j|jkr,|jddjrdSt|jddjr(|jdjr*dSdSdS|jSrS)r<r?rPrrrTr/r/r0rs   zlog._eval_is_algebraiccC |jdjSrvr?rLr-r/r/r0rrWzlog._eval_is_extended_realcCs|jd}t|jt|jgSrv)r?rrrrP)r.rSr/r/r0rs zlog._eval_is_complexcCs|jd}|jr dS|jSNrF)r?rPrMrNr/r/r0rOs zlog._eval_is_finitecC|jddjSNrr1rVr-r/r/r0rrDzlog._eval_is_extended_positivecCrXrY)r?rPr-r/r/r0rVrDzlog._eval_is_zerocCrXrY)r?is_extended_nonnegativer-r/r/r0_eval_is_extended_nonnegativerDz!log._eval_is_extended_nonnegativerc sddlm}ddlm}|}|st|}|jd|kr|S|jd}tdtd} } || || } | durY| | | | } } | dkrY| |sY| |sYt| | |} | Sdd} z| |\} }|j |||d}Wn&t t t fy|j ||d}|jrd 7|j ||d}|jsYnw|r||r|j||d tj} }n#z | |\} }Wnt t t fy||tj} }Ynwt|| ||d }|tr||}t||r|| ||\}}|jsRt| ||}|}td d d d d d d d d d }|jdi|}| s4|r4|| t| jdi|}n||t|jdi|}||krI|S||||Sfdd}i}t|D]}| ||\}}||tj|||<q_tj } i}|}| |krtj!|  | }|D]}||tj|||||<q|||}| tj 7} | |kst| ||}|D] }|||||7}q|dkr|jd"||}| j#r| j$rt%|dkr|dt&tj'8}||||S)Nrrrkrc Sstjtj}}t|D]/}||r7|\}}||kr6z||WSty5|tjfYSwq ||9}q ||fSr*) rr;rrrhasr@leadtermr6)rrarr+r>rr/r/r0 coeff_exps    z$log._eval_nseries..coeff_exprr1rTF) rr4mul power_exp power_base multinomialbasicr=r>csNi}t||D]\}}||}|kr$||tj||||||<q|Sr*)rrArr)d1d2r}e1e2exr r/r0r`s"zlog._eval_nseries..mulrr/)(rrrrr4r?rmatchr]r^rr6rr rrrrrr(rrr+rhrrdictr:rrrrAr;rdirr+r9r!rr) r.rarrrrr_logxrr\rr r_rr\rRr_dr}_reslogflagsr,r`ptermsrco1rgrpkrrir/r r0rs        "  "   zlog._eval_nseriescCs|jd}tdddd}|dkrd}||||}z |j||dd\}}Wnty=|j|||d} t| YSw||rY||||}|dkrUt d|t|S|t j krm|t j krm|t j j||dSt||t|} |durt|n|}| ||7} |j rt|dkrdd lm} t||D] \} } | jr| d krnq| d kr| |\}}| d tt j| t| d7} | S) NrrT)realpositiver1)rrrr) Heavisider2)r?togetherrrr^r6rr4r]r rr;rr9r!'sympy.functions.special.delta_functionsrw enumeratelseriesr+as_coeff_exponentrr)r.rarrrrrScr]rr}rwr{rrrnr/r/r0r7s>      "zlog._eval_as_leading_termrmr*r#r$rv) rnrorprtTupler__annotations__rrrrrsrr7r%rr@r&rrrFrNrrTrrrrOrrVr[rrr/r/r/r0r4]s4  !      9     jr4cs|eZdZdZeejddd ejfZe dddZ dd d Z d d Z d dZ ddZdddZdfdd ZddZZS)LambertWa The Lambert W function $W(z)$ is defined as the inverse function of $w \exp(w)$ [1]_. Explanation =========== In other words, the value of $W(z)$ is such that $z = W(z) \exp(W(z))$ for any complex number $z$. The Lambert W function is a multivalued function with infinitely many branches $W_k(z)$, indexed by $k \in \mathbb{Z}$. Each branch gives a different solution $w$ of the equation $z = w \exp(w)$. The Lambert W function has two partially real branches: the principal branch ($k = 0$) is real for real $z > -1/e$, and the $k = -1$ branch is real for $-1/e < z < 0$. All branches except $k = 0$ have a logarithmic singularity at $z = 0$. Examples ======== >>> from sympy import LambertW >>> LambertW(1.2) 0.635564016364870 >>> LambertW(1.2, -1).n() -1.34747534407696 - 4.41624341514535*I >>> LambertW(-1).is_real False References ========== .. [1] https://en.wikipedia.org/wiki/Lambert_W_function rFrXNcCsV|tjkr ||S|durtj}|jro|jrtjS|tjur!tjS|dtjkr+tjS|td dkr9td S|dtdkrEtdS|tj dkrUtjtjdS|t dtjkratjS|tj uritj S|jrotjSt |jrz|jrztj S|tjur|tj dkrtj tjdS|dtjkrtjS|dt dkrt d SdSdS)Nrrr1rx)rrrPrr;rr4rrr+rrrUr)rrar\r/r/r0rsF       z LambertW.evalr1cCsr|jd}t|jdkr|dkrt||dt|Sn|jd}|dkr4t|||dt||St||)z? Return the first derivative of this function. rr1)r?rrr )r.r6rar\r/r/r0rs   zLambertW.fdiffcCs|jd}t|jdkrtj}n|jd}|jr.|dtjjr"dS|dtjjr,dSdS|djrO|jr@|dtjjr@dS|jsK|dtjj rMdSdSt |jr`t |djrb|j rddSdSdSdSrS) r?rrrrPrrrGr9r7rr~)r.rar\r/r/r0rs*   zLambertW._eval_is_extended_realcCrUrv)r?rMr-r/r/r0rOrWzLambertW._eval_is_finitecCsF|j|j}|j|jkr t|jdjr|jdjrdSdSdS|jSrW)r<r?rrPrrTr/r/r0rs  zLambertW._eval_is_algebraicrcCsFt|jdkr!|jd}||d}|js||S||SdS)Nr1r)rr?rr(rPr<r)r.rarrrrr/r/r0rs   zLambertW._eval_as_leading_termc st|jdkrXddlm}ddlm}|jdj|||dj||d}d}|jr-|j }|||dkrLt fddt d|||D} t | } nt j} | ||||St|||S) Nr1rrrrrcs@g|]}tj |dt||dt|d|qS)r1r)rr;rr)r`r\rr/r0r s z*LambertW._eval_nseries..)rr?rrrrrcompute_leading_termrr+rrrrrsuperr) r.rarrrrrltlterRrrr0rs     zLambertW._eval_nseriescCs4|jd}t|jdkr|jSt|j|jdjgSrY)r?rrPr)r.rar/r/r0rVs zLambertW._eval_is_zeror*rmrvr$)rnrorprrrrrrrsr%rrrrOrrrrV __classcell__r/r/rr0rds"  % rN)C itertoolsrtypingrrsympy.core.exprr sympy.corersympy.core.addrsympy.core.cachersympy.core.functionrr r r r r rrsympy.core.logicrrrsympy.core.mulrsympy.core.numbersrrrrrsympy.core.parametersrsympy.core.powerrsympy.core.singletonrsympy.core.symbolrr(sympy.functions.combinatorial.factorialsrrrr r!r"r#(sympy.functions.elementary.miscellaneousr$ sympy.ntheoryr%r&sympy.ntheory.factor_r'sympy.polys.polytoolsr(r)rurr+r/r4rr/r/r/r0sB     (        fIp