o 3Ihr@sddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z mZmZmZmZmZddlmZmZmZddlmZdd lmZmZmZmZdd lmZdd l m!Z!dd l"m#Z#dd l$m%Z%m&Z&ddl'm(Z(ddl)m*Z*ddl+m,Z,m-Z-m.Z.m/Z/m0Z0ddl1m2Z2ddl3m4Z4m5Z5ddl6m7Z7Gddde Z8Gddde8Z9GdddeZ:Gddde8e:dZ;ddZed#d$Z?d%S)&) annotations)product)Add)cacheit)Expr)DefinedFunctionArgumentIndexError expand_log expand_mul FunctionClass PoleErrorexpand_multinomialexpand_complex) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRationalpiI)global_parameters)Pow)S)WildDummy)sympify) factorial)arg unpolarifyimreAbs)sqrt) multiplicity perfect_power) factorintc@seZdZdZejfZeddZdddZ ddZ ed d Z d d Z d dZ ddZddZddZddZddZddZddZdS)ExpBaseTcCs|jjSN)expkindselfr-f/home/air/sanwanet/gpt-api/venv/lib/python3.10/site-packages/sympy/functions/elementary/exponential.pyr*(sz ExpBase.kindcCtS)z= Returns the inverse function of ``exp(x)``. logr,argindexr-r-r.inverse,zExpBase.inversecCsP|js|tjfS|j}|j}|s| js|}|r#tj|| 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) )is_commutativerOner) is_negativecould_extract_minus_signfunc)r,r)neg_expr-r-r.as_numer_denom2s   zExpBase.as_numer_denomcCs |jdS)z7 Returns the exponent of the function. r)argsr+r-r-r.r)Ls z ExpBase.expcCs|dt|jfS)z7 Returns the 2-tuple (base, exponent). r/)r;rr>r+r-r-r. as_base_expSszExpBase.as_base_expcC||jSr()r;r)adjointr+r-r-r. _eval_adjointYzExpBase._eval_adjointcCr@r()r;r) conjugater+r-r-r._eval_conjugate\rCzExpBase._eval_conjugatecCr@r()r;r) transposer+r-r-r._eval_transpose_rCzExpBase._eval_transposecCs.|j}|jr|jr dS|jrdS|jrdSdSNTF)r) is_infiniteis_extended_negativeis_extended_positive is_finiter,rr-r-r._eval_is_finitebszExpBase._eval_is_finitecCsJ|j|j}|j|jkr"|jj}|rdS|jjrt|r dSdSdS|jSrH)r;r>r)is_zero is_rationalr)r,szr-r-r._eval_is_rationalls  zExpBase._eval_is_rationalcCs |jtjuSr()r)rNegativeInfinityr+r-r-r. _eval_is_zerow 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-r.rYzs 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-r. z1ExpBase._eval_expand_power_exp..) sympy.concrete.productsr]sympy.concrete.summationsr^r>is_Addr7rfromiter isinstancer;functionlimits)r,hintsr]r^rr-r+r._eval_expand_power_exps     zExpBase._eval_expand_power_expNr/)__name__ __module__ __qualname__ unbranchedrComplexInfinity_singularitiespropertyr*r5r=r)r?rBrErGrNrSrUrYrkr-r-r-r.r'#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-r. _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-r.rxs zexp_polar._eval_evalfcCs||jd|Sru)r;r>)r,rZr-r-r.rYszexp_polar._eval_powercCs|jdjrdSdS)NrT)r>is_extended_realr+r-r-r._eval_is_extended_reals z exp_polar._eval_is_extended_realcCs"|jddkr |tjfSt|Sru)r>rr8r'r?r+r-r-r.r?s  zexp_polar.as_base_expN) rmrnro__doc__is_polar is_comparablervrxrYr~r?r-r-r-r.rts% rtc@seZdZddZdS)ExpMetacCs&t|jjvrdSt|to|jtjuS)NT)r) __class____mro__rgrbaserExp1)clsinstancer-r-r.__instancecheck__s zExpMeta.__instancecheck__N)rmrnrorr-r-r-r.rs rcseZdZdZd+ddZddZeddZed d Z e e d d Z d,ddZ fddZddZddZddZddZd-ddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*ZZS).r)a9 The exponential function, :math:`e^x`. Examples ======== >>> from sympy import exp, I, pi >>> from sympy.abc import x >>> exp(x) exp(x) >>> exp(x).diff(x) exp(x) >>> exp(I*pi) -1 Parameters ========== arg : Expr See Also ======== log r/cCs|dkr|St||)z@ Returns the first derivative of this function. r/)rr3r-r-r.fdiffs z exp.fdiffcCsddlm}m}|jd}|jr^ttj}||| fvrtjS| t t}|r`|| d|rb|| |r;tj S|||rEtjS|| |tjrRt S|||tjrdtSdSdSdSdS)Nr)askQ)sympy.assumptionsrrr>is_MulrrInfinityNaNas_coefficientrintegerevenr8odd NegativeOneHalf)r, assumptionsrrrIoocoeffr-r-r. _eval_refines*  zexp._eval_refinecCsdddlm}ddlm}ddlm}ddlm}t||r!| St j r*t t j|S|jrU|t jur5t jS|jr;t jS|t jurCt jS|t jurKt jS|t jurSt jSnT|t jur]t jSt|trg|jdSt||rw|t |jt |jSt||r||S|jrO|tt}|rd|j r|j!rt jS|j"rt j#S|t j$j!rt S|t j$j"rtSn|j%r|d}|dkr|d8}||kr||ttS|&\}}|t jt jfvr|j'r|t jur| }t(|jr|t jurt jSt(|j)rt*|t jurt jSt(|j+rt jSdS|gd} } t,-|D](} || } t| tr6| dur3| jd} qdS| j.rA| /| qdS| rM| t,| SdS|j0rg} g}d}|jD]:}|t jurk|/|q\||}t||r|jd|kr|/|jdd }q\|/|q\| /|q\| s|rt,| |t1|dd S|jrt jSdS) Nr AccumBounds) MatrixBaseSetExpr logcombinerr/FTrW)2sympy.calculusrsympy.matrices.matrixbasersympy.sets.setexprrsympy.simplify.simplifyrrgr)r exp_is_powrrr is_NumberrrOr8rrTZerorqr2r>minmax _eval_funcrrrr is_integeris_evenis_oddrr is_Rational as_coeff_Mul is_numberr! is_positiver r9r make_argsrappendrer)rrrrrrrncoefftermscoeffslog_termtermterm_outadd argchangedanewar-r-r.evals                              zexp.evalcCstjS)z? Returns the base of the exponential function. )rrr+r-r-r.r}szexp.basecGsT|dkrtjS|dkrtjSt|}|r"|d}|dur"|||S||t|S)zJ Calculates the next term in the Taylor series expansion. rN)rrr8rr)nr`previous_termspr-r-r. 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,deeprjrrr!r r-r-r.rszexp.as_real_imagcs|jrt|jt|j}n |tjur|jrt}t|ts"|tjur1dd}t |||||S|turA|jsA||j ||St ||S)NcSs&|jst|trt|ddiS|S)NrXF)is_Powrgr)rr?)rr-r-r.s z exp._eval_subs..) rr)r2rrr is_Functionrgr _eval_subs_subssuper)r,oldnewfrr-r.rszexp._eval_subscCsB|jdjrdS|jdjrtd t|jdt}|jSdS)NrTr)r>r} is_imaginaryrrrrr,arg2r-r-r.r~s  zexp._eval_is_extended_realcCsdd}t||jdS)Ncss|jV|jVdSr() is_complexrJrr-r-r.complex_extended_negatives z7exp._eval_is_complex..complex_extended_negativer)rr>)r,rr-r-r._eval_is_complexszexp._eval_is_complexcCsD|jttjr dSt|jjr|jjrdS|jtjr dSdSdSrH)r)rrrPrrO is_algebraicr+r-r-r._eval_is_algebraics  zexp._eval_is_algebraiccCs>|jjr |jdtjuS|jjrt |jdt}|jSdSru) r)r}r>rrTrrrrrr-r-r._eval_is_extended_positives zexp._eval_is_extended_positiverc sddlmddlm}ddlm}ddlm}ddlm }|j } | j |||d} | j r0d| S|| |d} | tjurD||||S| tjurK|S| jrTtd |tfd d | jDrb|Std } |} z|| j||d |}Wn ttfyd}Ynw|r|dkr|||} t | | | }t | || | | }|dur|t|ini}|||kr|S|r|dkr||| | ||||d|7}n ||| | ||7}|}||ddd}dd}td|gd}|tj|t tj|}|S)Nrsignceiling)limitOrderpowsimprlogxr/Cannot expand %s around 0c3s|]}t|VqdSr()rg)r_rrr-r.rarbz$exp._eval_nseries..trTr)rcombinecSs|jo|jdvS)N))rq)r`r-r-r.r sz#exp._eval_nseries..w) properties)!$sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimprr) _eval_nseriesis_OrderremoveOrrTrrIr anyr>ras_leading_termgetnNotImplementedError_taylorsubsr2rrreplacerr)r,r`rrcdirrrrrr arg_seriesarg0rntermscf exp_seriesrrep simpleratrr-rr.rsR           (zexp._eval_nseriescCsNg}d}t|D]}|||jd|}|j||d}||qt|S)Nrr)rangerr>nseriesrrr)r,r`rlgrzr-r-r.rs z exp._taylorcCsddlm}|jdj||d}||d}|tjur tjSt||r5t |tj kr1t | St |S|tjur@| |d}|j durIt |Std|)NrrrFr)sympy.calculus.utilrr>cancelrrrrrgr!rr)rrIr )r,r`rrrrrr-r-r._eval_as_leading_terms         zexp._eval_as_leading_termcKs0ddlm}|t|tdt|t|S)Nr)rr)rrrr)r,rkwargsrr-r-r._eval_rewrite_as_sin. $zexp._eval_rewrite_as_sincKs0ddlm}|t|t|t|tdS)Nr)rr)rrrr)r,rrrr-r-r._eval_rewrite_as_cos2rzexp._eval_rewrite_as_coscKs,ddlm}d||dd||dS)Nr)tanhr/r)%sympy.functions.elementary.hyperbolicr)r,rrrr-r-r._eval_rewrite_as_tanh6s  zexp._eval_rewrite_as_tanhcKs|ddlm}m}|jr4|tt}|r6|jr8|t||t|}}t||s:t||s<|t|SdSdSdSdSdS)Nr)rr) rrrrrrrrrg)r,rrrrrcosinesiner-r-r._eval_rewrite_as_sqrt:s  zexp._eval_rewrite_as_sqrtcKs@|jrdd|jD}|rt|djd||dSdSdS)NcSs(g|]}t|trt|jdkr|qSrl)rgr2lenr>)r_rr-r-r. Es(z,exp._eval_rewrite_as_Pow..r)rr>rr)r,rrlogsr-r-r._eval_rewrite_as_PowCs zexp._eval_rewrite_as_PowrlTr)rmrnrorrr classmethodrrsr staticmethodrrrrr~rrrrrrrrrr r$ __classcell__r-r-rr.r)s2   i     /  r)) metaclasscCsN|jtdd\}}|dkr|jr||fS|t}|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_independentris_realr)exprr_i_r-r-r.match_real_imagJs  r2c@seZdZUdZded<ejejfZd*ddZ d*ddZ e d+d d Z e ed d Zd,ddZddZd,ddZddZddZddZddZddZdd Zd!d"Zd#d$Zd-d&d'Zd(d)Zd S).r2a The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp z tuple[Expr]r>r/cCs |dkr d|jdSt||)z? Returns the first derivative of the function. r/r)r>rr3r-r-r.rs z log.fdiffcCr0)zC Returns `e^x`, the inverse function of `\log(x)`. )r)r3r-r-r.r5r6z log.inverseNcCsddlm}ddlm}t|}|dur`t|}|dkr&|dkr#tjStjSzt||}|r=|t |||t |WSt |t |WSt yNYnw|tj ur\||||S||S|j r|j ritjS|tjurqtjS|tjurytjS|tjurtjS|tjurtjS|jr|jdkr||j S|jr|jtj ur|jjr|jSt|tr|jjr|jSt|tr|jjrt|j\}}|r|jr|dt;}|tkr|dt8}|t|tddSn.r-)%sympy.concreter^r]getr!r>r r; is_Integerr%r&keyssumitemsrr2rrrrrrgr_eval_expand_logr9rrrrrr)r}ris_nonpositiverrhri)r,rrjr^r]r=r>rrlogargrr/nonposr`rr[r\r-r-r.rF$sp                  zlog._eval_expand_logcKsddlm}m}m}t|jdkr||j|jfi|S|||jdfi|}|dr3||}||dd}t||g|ddS) Nr)r simplifyinversecombinerr5Tr3measure)key)rr rJrKr!r>r;r)r,rr rJrKr/r-r-r._eval_simplify]s 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)) rr2Fcomplex)r>rr"rrrrAr2)r,rrjsargsarg_abssarg_argr-r-r.rhs    zlog.as_real_imagcCs^|j|j}|j|jkr,|jddjrdS|jdjr(t|jddjr*dSdSdS|jSNrr/TF)r;r>rOrPrr,rQr-r-r.rSs   zlog._eval_is_rationalcCs^|j|j}|j|jkr,|jddjrdSt|jddjr(|jdjr*dSdSdS|jSrS)r;r>rOrrrTr-r-r.rs   zlog._eval_is_algebraiccC |jdjSrur>rKr+r-r-r.r~rVzlog._eval_is_extended_realcCs|jd}t|jt|jgSru)r>rrrrO)r,rRr-r-r.rs zlog._eval_is_complexcCs|jd}|jr dS|jSNrF)r>rOrLrMr-r-r.rNs zlog._eval_is_finitecC|jddjSNrr/rVr+r-r-r.rrCzlog._eval_is_extended_positivecCrXrY)r>rOr+r-r-r.rUrCzlog._eval_is_zerocCrXrY)r>is_extended_nonnegativer+r-r-r._eval_is_extended_nonnegativerCz!log._eval_is_extended_nonnegativerc$ sddlm}ddlm}ddlm}|jd|kr#|dur!t|S|S|jd}|ddd} |dkr4d}|||| } t d t d } } | | | | } | dur| | | | } } | dkr| | s| | s|durs| t|n| |} | t| | t|7} | Sd d }z | j | |dd \}}WnLt ttfy| j| |dd}|jrd7| j| |dd}|jsz |j | dd\}}Wnt y|j| ddtj}}YnwYnw| || |dj| |dd}| tr||}t||r|||| \}}|durt|n|}|js{t||t|||}|}dddddddddd }|jdi|}|s]|r]|| t| jdi|}n||t|jdi|}||krr|S||||Sfdd}i}t|D]}||| \}}||tj|||<qtj } i}|}| |krtj!|  | } |D]}!||!tj| ||!||!<q|||}| tj 7} | |kst||t|||}|D]}!|||!| |!7}q|j"rAt#| dkrAddl$m%}"t&| '| D]\}#}|j(r!|#dkr#nq|#dkrA|)| \} }|dt*t+|"t#|  d7}|| ||}||||S)Nrrr)rrTpositiver/krc Sstjtj}}t|D]/}||r7|\}}||kr6z||WSty5|tjfYSwq ||9}q ||fSr() rr8rrrhasr?leadtermr5)rr`rr)r>rr-r-r. coeff_exps    z$log._eval_nseries..coeff_exprr)rrr)rF) rr2mul power_exp power_base multinomialbasicr=r>csNi}t||D]\}}||}|kr$||tj||||||<q|Sr()rrArr)d1d2r|e1e2exrr-r.rcs"zlog._eval_nseries..mul Heavisider-),rrrrsympy.core.symbolrr>r2rrmatchr_r`r5rr rrrrrrrr)rgrrrr:rrrAr8rr9r 'sympy.functions.special.delta_functionsrn enumeratelseriesr.as_coeff_exponentrr)$r,r`rrrrrrrrrRr^rr rarr[rQr_dr|_reslogflagsr/rcptermsrco1rjrpkrrlrnrzr-rr.rs      &   "  "     zlog._eval_nseriescCs|jd}tddd}|dkrd}||||}z |j||dd\}}Wnty<|j|||d} t| YSw||rX||||}|dkrTt d|t|S|t j krl|t j krl|t j j||dSt||t|} |dur~t|n|}| ||7} |j rt|dkrdd lm} t||D] \} } | jr| d krnq| d kr| |\}}| d tt| t| d7} | S) NrrTr\r/rbrrrmrorp)r>togetherrrr`r5rr2r_r rr8rr9r rsrnrtrur.rvrr)r,r`rrrrrRcr\rr|rnrzrrrwr-r-r.r's>        zlog._eval_as_leading_termrlr(r%r&)rmrnror__annotations__rrrqrrrr5r'rr(rrrFrNrrSrr~rrNrrUr[rrr-r-r-r.r2_s0 !    }  9      tr2cszeZdZdZeejddd ejfZe dddZ dd d Z d d Z d dZ ddZddZdfdd 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 rFrWNcCs>|tjkr ||S|durtj}|jrf|jrtjS|tjur!tjS|dtjkr+tjS|td dkr9td S|dtdkrEtdS|t dkrRttdS|t dtjkr^tjS|tj urftj St |jrq|jrqtj S|tjur|t dkrt tdS|dtjkrtjS|dt dkrt d SdSdS)Nrrr/rp)rrrOrr8rr2rrr)rrrTr)rr`r^r-r-r.rysB        z LambertW.evalr/cCsr|jd}t|jdkr|dkrt||dt|Sn|jd}|dkr4t|||dt||St||)z? Return the first derivative of this function. rr/)r>r!rr)r,r4r`r^r-r-r.rs   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>r!rrrOrrrGr9r6rr})r,r`r^r-r-r.r~s*   zLambertW._eval_is_extended_realcCrUru)r>rLr+r-r-r.rNrVzLambertW._eval_is_finitecCsF|j|j}|j|jkr t|jdjr|jdjrdSdSdS|jSrW)r;r>rrOrrTr-r-r.rs  zLambertW._eval_is_algebraiccCsFt|jdkr!|jd}||d}|js||S||SdS)Nr/r)r!r>rrrOr;r)r,r`rrrrr-r-r.rs   zLambertW._eval_as_leading_termrc st|jdkrXddlm}ddlm}|jdj|||dj||d}d}|jr-|j }|||dkrLt fddt d|||D} t | } nt j} | ||||St|||S) Nr/rrrrrcs@g|]}tj |dt||dt|d|qS)r/r)rr8rr)r_r^rr-r.r"s z*LambertW._eval_nseries..)r!r>rrrrrrrr)rrr rrrr) r,r`rrrrrltlterQrrr.rs     zLambertW._eval_nseriescCs4|jd}t|jdkr|jSt|j|jdjgSrY)r>r!rOr)r,r`r-r-r.rUs zLambertW._eval_is_zeror(rlr&)rmrnrorrrrrqrrr'rrr~rNrrrrUr)r-r-rr.rTs"  #rcCsHitdtddtdtddtdtdtdtdtddtdtdtddtdttddtdttdddtdttddtddtdtddtdtdtdttddtdtddttddttddtdtdttddtddtddtd td td dttddtd tddtddttdd tdtd dtdtdttdd dtdtd dtddtdtd dtdttdd dtddtdttdd iS) Nrr/rrorrr )r#rrr-r-r-r.r8sL (. $  *  *"."r8N)@ __future__r itertoolsrsympy.core.addrsympy.core.cachersympy.core.exprrsympy.core.functionrrr r r r r rsympy.core.logicrrrsympy.core.mulrsympy.core.numbersrrrrsympy.core.parametersrsympy.core.powerrsympy.core.singletonrrqrrsympy.core.sympifyr(sympy.functions.combinatorial.factorialsrrrrr r!r"(sympy.functions.elementary.miscellaneousr# sympy.ntheoryr$r%sympy.ntheory.factor_r&r'rtrr)r2r2rr8r-r-r-r.sD    (        hIqx