U -eƦ@sddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z mZmZmZmZmZmZddlmZmZmZddlmZdd lmZmZmZmZmZdd l m!Z!dd l"m#Z#dd l$m%Z%dd l&m'Z'm(Z(ddl)m*Z*ddl+m,Z,ddl-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4ddl5m6Z6m7Z7ddl8m9Z9Gddde Z:Gddde:Z;GdddeZGdd d e Z?Gd!d"d"e Z@ed#d$ZAd%S)&)product)Tuple)Add)cacheit)Expr)FunctionArgumentIndexError expand_log expand_mul FunctionClass PoleErrorexpand_multinomialexpand_complex) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRationalpiI ImaginaryUnit)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.g/var/www/html/Darija-Ai-Train/env/lib/python3.8/site-packages/sympy/functions/elementary/exponential.pyr+(sz ExpBase.kindcCstS)z= Returns the inverse function of ``exp(x)``. logr-argindexr.r.r/inverse,szExpBase.inversecCs@|j}|j}|s | js |}|r6tj|| 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*Zneg_expr.r.r/as_numer_denom2s zExpBase.as_numer_denomcCs |jdS)z7 Returns the exponent of the function. r)argsr,r.r.r/r*Jsz ExpBase.expcCs|dt|jfS)z7 Returns the 2-tuple (base, exponent). r0)r9rr;r,r.r.r/ as_base_expQszExpBase.as_base_expcCs||jSr))r9r*Zadjointr,r.r.r/ _eval_adjointWszExpBase._eval_adjointcCs||jSr))r9r* conjugater,r.r.r/_eval_conjugateZszExpBase._eval_conjugatecCs||jSr))r9r*Z transposer,r.r.r/_eval_transpose]szExpBase._eval_transposecCs.|j}|jr |jrdS|jr dS|jr*dSdSNTF)r* is_infiniteis_extended_negativeis_extended_positive is_finiter-rr.r.r/_eval_is_finite`szExpBase._eval_is_finitecCsH|j|j}|j|jkr>|jj}|r(dS|jjrDt|rDdSn|jSdSrA)r9r;r*is_zero is_rationalr)r-szr.r.r/_eval_is_rationaljs  zExpBase._eval_is_rationalcCs |jtjkSr))r*rNegativeInfinityr,r.r.r/ _eval_is_zerouszExpBase._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/rQxs zExpBase._eval_powerc szddlm}ddlm}jd}|jrH|jrHtfdd|jDSt ||rp|jrp| |j f|j S |S)Nr)Product)Sumc3s|]}|VqdSr))r9).0xr,r.r/ sz1ExpBase._eval_expand_power_exp..) Zsympy.concrete.productsrUZsympy.concrete.summationsrVr;is_AddZis_commutativerZfromiter isinstancer9functionlimits)r-hintsrUrVrr.r,r/_eval_expand_power_exp~s    zExpBase._eval_expand_power_expN)r0)__name__ __module__ __qualname__Z unbranchedrComplexInfinity_singularitiespropertyr+r5r:r*r<r=r?r@rGrLrNrQr_r.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_AbscCszt|jd}z|t kp |tk}Wntk r<d}YnX|rF|St|jd|}|dkrvt|dkrvt|S|S)z. Careful! any evalf of polar numbers is flaky rT)r!r;r TypeErrorr* _eval_evalfr")r-precibadresr.r.r/rjs zexp_polar._eval_evalfcCs||jd|Srg)r9r;)r-rRr.r.r/rQszexp_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|Srg)r;rr8r(r<r,r.r.r/r<s zexp_polar.as_base_expN) r`rarb__doc__is_polar is_comparablerhrjrQrpr<r.r.r.r/rfs%rfc@seZdZddZdS)ExpMetacCs&t|jjkrdSt|to$|jtjkS)NT)r* __class____mro__r[rbaserExp1)clsinstancer.r.r/__instancecheck__s zExpMeta.__instancecheck__N)r`rarbr{r.r.r.r/rtsrtc@seZdZdZd,ddZddZeddZed d Z e e d d Z d-ddZ ddZddZddZddZddZd.ddZddZd/d d!Zd"d#Zd$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 r0cCs|dkr |St||dS)z@ Returns the first derivative of this function. r0N)rr3r.r.r/fdiffsz exp.fdiffcCsddlm}m}|jd}|jrttj}||| fkr>tjS| t t}|r|| d|r|| |rvtj S|||rtjS|| |tjrt S|||tjrtSdS)Nr)askQ)Zsympy.assumptionsr}r~r;is_MulrrInfinityNaNas_coefficientrintegerZevenr8Zodd NegativeOneHalf)r-Z assumptionsr}r~rZIoocoeffr.r.r/ _eval_refines"  zexp._eval_refinecCs~ddlm}ddlm}ddlm}ddlm}t||rB| St j rTt t j|S|jr|t jkrjt jS|jrvt jS|t jkrt jS|t jkrt jS|t jkrt jSn|t jkrt jSt|tr|jdSt||r|t |jt |jSt||r||S|jr|tt}|rd|j rr|j!r>> 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-deepr^rrr"r!r.r.r/rszexp.as_real_imagcCs|jrt|jt|j}n|tjkr0|jr0t}t|tsD|tjkrbdd}t |||||S|tkr|js||j ||St |||S)NcSs&|jst|tr"t|ddiS|S)NrPF)is_Powr[r*rr<)rr.r.r/s z exp._eval_subs..) rr*r2rwrrxZ is_Functionr[r _eval_subs_subsr)r-oldnewfr.r.r/rszexp._eval_subscCsB|jdjrdS|jdjr>td t|jdt}|jSdS)NrTr)r;ro is_imaginaryrrrrr-Zarg2r.r.r/rps   zexp._eval_is_extended_realcCsdd}t||jdS)Ncss|jV|jVdSr)) is_complexrCrr.r.r/complex_extended_negativesz7exp._eval_is_complex..complex_extended_negativer)rr;)r-rr.r.r/_eval_is_complexszexp._eval_is_complexcCs@|jttjrdSt|jjr<|jjr,dS|jtjr|jjr|jdtjk S|jjr:t |jdt}|jSdSrg) r*ror;rrMrrrrrr.r.r/_eval_is_extended_positives zexp._eval_is_extended_positiverc sddlmddlm}ddlm}ddlm}ddlm }|j } | j |||d} | j r`d| S|| |d} | tjkr||||S| tjkr|Stfd d | jDr|Std } |} z|| j||d |}Wnttfk rd}YnX|r|dkr|||} t | | | }t | || | | }|dk rR|t|ini}|||krj|S|r|dkr||| | ||||d|7}n||| | ||7}|}||d dd}dd}td|gd}|tj|ttj|}|S)Nrsignceiling)limitOrderpowsimprlogxr0c3s|]}t|tfVqdSr))r[r)rWrrr.r/rYsz$exp._eval_nseries..trTr*rcombinecSs|jo|jdkS)N))rq)rXr.r.r/rz#exp._eval_nseries..w) properties) $sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrZsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimprr* _eval_nseriesis_OrderremoveOrrMranyr;ras_leading_termgetnNotImplementedErrorr _taylorsubsr2rrreplacerr)r-rXrrcdirrrrrrZ arg_seriesarg0rZntermscfZ exp_seriesrrepZ simpleratrr.rr/rsL         (zexp._eval_nseriescCsNg}d}t|D]4}|||jd|}|j||d}||qt|S)Nrr)rangerr;nseriesrrr)r-rXrlgrlr.r.r/r s z exp._taylorNcCsddlm}|jdj||d}||d}|tjkr@tjSt||rjt |tj krbt | St |S|tjkr| |d}|j dkrt |Std|dS)NrrrFCannot expand %s around 0)Zsympy.calculus.utilrr;cancelrrrrr[r"rr*rrBr )r-rXrrrrrr.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*s zexp._eval_rewrite_as_sincKs0ddlm}|t|t|t|tdS)Nr)rr)rrrr)r-rrrr.r.r/_eval_rewrite_as_cos.s zexp._eval_rewrite_as_coscKs,ddlm}d||dd||dS)Nr)tanhr0r)Z%sympy.functions.elementary.hyperbolicr)r-rrrr.r.r/_eval_rewrite_as_tanh2s zexp._eval_rewrite_as_tanhcKslddlm}m}|jrh|tt}|rh|jrh|t||t|}}t||sht||sh|t|SdS)Nr)rr) rrrrrrrrr[)r-rrrrrZcosineZsiner.r.r/_eval_rewrite_as_sqrt6s zexp._eval_rewrite_as_sqrtcKs<|jr8dd|jD}|r8t|djd||dSdS)NcSs(g|] }t|trt|jdkr|qS)r0)r[r2lenr;)rWrr.r.r/ As z,exp._eval_rewrite_as_Pow..r)rr;rr)r-rrZlogsr.r.r/_eval_rewrite_as_Pow?szexp._eval_rewrite_as_Pow)r0)T)r)Nr)r`rarbrqr|r classmethodrrerw staticmethodrrrrrprrrrrrrrrrrr.r.r.r/r*s0  i     -  r*) metaclasscCsR|jtdd\}}|dkr(|jr(||fS|t}|rJ|jrJ|jrJ||fSdSdS)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. TZas_Addr)NNN)as_independentris_realr)exprr_i_r.r.r/match_real_imagFs  rc@seZdZUdZeeed<ejej fZ d+ddZ d,ddZ e d-d d Zd d Zeed dZd.ddZddZd/ddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd0d'd(Zd1d)d*ZdS)2r2a The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp r;r0cCs$|dkrd|jdSt||dS)z? Returns the first derivative of the function. r0rN)r;rr3r.r.r/r|sz log.fdiffcCstS)zC Returns `e^x`, the inverse function of `\log(x)`. )r*r3r.r.r/r5sz log.inverseNcCsddlm}ddlm}t|}|dk rt|}|dkrL|dkrFtjStjSzBt||}|rz|t |||t |WSt |t |WSWnt k rYnX|tj k r||||S||S|j r@|j rtjS|tjkrtjS|tjkrtjS|tjkrtjS|tjkr tjS|jr@|jdkr@||j S|jrf|jtj krf|jjrf|jSt|tr|jjr|jSt|tr|jjrt|j\}}|rh|jrh|dt;}|tkr|dt8}|t|tddSn|t|t rt!|jSt||rR|j"j#r.|t |j"t |j$S|j"j rJ|tjt |j$StjSnt||rh|%|S|jr|j&rtt|| S|tjkrtjS|tj krtjS|j rtjS|j's<|(t}|dk r<|tjkrtjS|tjkrtjS|jr<|j)r"tttj*||St ttj*|| S|jr|j+r|j,tdd\}} |j&rv|d 9}| d 9} t| dd} | j,td d\}}|(t}|j-r|r|j-r|j-r|j r|j#rtttj*|||S|j&rt ttj*||| Sndd l.m/} ||0} | 0} t1} | | kr| |t2| }|j#rl||t| | S||t| | tSnP| | kr| |t2| }|j#r||t| |  S||tt| | SdS) Nrrrr0rFrrrT)ratsimp)3rrrrrrrrcr%r2 ValueErrorrxrrHr8rrrMrrrrrwr*ror[rrrsrr rrfr rrrrr6rZris_nonnegativerrrrZsympy.simplifyrr_log_atan_tabler#)ryrrwrrrrrrZarg_rrt1Z atan_tablemodulusr.r.r/rs                                  zlog.evalcCs |tjfS)zE Returns this function in the form (base, exponent). )rr8r,r.r.r/r<szlog.as_base_expcGsddlm}|dkrtjSt|}|dkr.|S|rb|d}|dk rb|| |||ddddSdd |d ||d|dS) zV Returns the next term in the Taylor series expansion of `\log(1+x)`. rrrNr0Tr*rr)rrrrr)rrXrrrr.r.r/rs  zlog.taylor_termTcKsnddlm}m}|dd}|dd}t|jdkrLt|j|j||dS|jd}|jrt |}d} d} |dk r|\}} ||} |rt |}|| krt d d | D} | dk r| | Sn|jrt|jt|jS|jrg} g} |jD]} |s| js| jrN|| }t|trB| || jf|n | |q| jrz|| }| || tjq| | qt| tt| S|jst|tr2|s|jjr|j js|jdjr|jdj!s|j jrd|j }|j}||}t|tr$t"||jf|St"||Sn2t||rd|sN|j#jrd|t|j#f|j$S||S) Nr)rVrUforceFfactorr)rr r0css|]\}}|t|VqdSr)r1)rWvalrr.r.r/rY9sz'log._eval_expand_log..)%Zsympy.concreterVrUgetrr;r r9Z is_Integerr&r'keyssumitemsrr2rrrrrrr[r_eval_expand_logr6rrrrrr*rorwis_nonpositiver r\r])r-rr^rVrUr r rrZlogargrrZnonposrXrrSrTr.r.r/r&sh             (    zlog._eval_expand_logcKs~ddlm}m}m}t|jdkr6||j|jf|S|||jdf|}|dr^||}||dd}t||g|ddS) Nr)r simplifyinversecombinerr5Trmeasure)key)rr rrrr;r9r)r-rr rrrr.r.r/_eval_simplify_s zlog._eval_simplifycKs~|jd}|r"|jdj|f|}t|}||kr<|tjfSt|}|ddrnd|d<t|j|f||fSt||fSdS)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)) rr2FcomplexN)r;rr#rrrrr2)r-rr^ZsargZsarg_absZsarg_argr.r.r/rjs   zlog.as_real_imagcCs\|j|j}|j|jkrR|jddjr,dS|jdjrXt|jddjrXdSn|jSdSNrr0TF)r9r;rHrIrr-rJr.r.r/rLs   zlog._eval_is_rationalcCs\|j|j}|j|jkrR|jddjr,dSt|jddjrX|jdjrXdSn|jSdSr)r9r;rHrrrr.r.r/rs   zlog._eval_is_algebraiccCs |jdjSrgr;rDr,r.r.r/rpszlog._eval_is_extended_realcCs|jd}t|jt|jgSrg)r;rrrrH)r-rKr.r.r/rs zlog._eval_is_complexcCs|jd}|jrdS|jSNrF)r;rHrErFr.r.r/rGs zlog._eval_is_finitecCs|jddjSNrr0rr,r.r.r/rszlog._eval_is_extended_positivecCs|jddjSr)r;rHr,r.r.r/rNszlog._eval_is_zerocCs|jddjSr)r;Zis_extended_nonnegativer,r.r.r/_eval_is_extended_nonnegativesz!log._eval_is_extended_nonnegativerc$ sddlm}ddlm}ddlm}|jd|krF|dkrBt|S|S|jd}|ddd} |dkrhd}|||| } t d t d } } | | | | } | dk r| | | | } } | dkr| | s| | s|dkr| t|n| |} | t| | t|7} | Sd d }z| j | |dd \}}Wnt ttfk r| j| |dd}|jrd7| j| |dd}q\z|j | dd\}}Wn0t k r|j| ddtj}}YnXYnX| || |dj| |dd}| tr ||}t||r|||| \}}|dkr>t|n|}|jst||t|||}|}dddddddddd }|jf|}|s|r|| t| jf|}n||t|jf|}||kr|S||||Sfdd}i}t|D]*}||| \}}||tj|||<qtj} i}|}| |krtj |  | } |D],}!||!tj| ||!}|!||!<qr|||}| tj7} qPt||t|||}|D]}!|||!| |!7}q|j"rt#| dkrddl$m%}"t&| '| D]"\}#}|j(r>|#dkr$qHq$|#dkr|)| \} }|dt*t+|"t#|  d7}|| ||}||||S)Nrrr)rrTZpositiver0krc Sstjtj}}t|D]`}||rp|\}}||krxz||WStk rl|tjfYSXq||9}q||fSr)) rr8rrrhasr<leadtermr)rrXrr*r rwr.r.r/ coeff_exps   z$log._eval_nseries..coeff_exprr)rrr)rF) rr2mulZ power_expZ power_baseZ multinomialbasicr r csNi}t||D]:\}}||}|kr||tj||||||<q|Sr))rrrr)Zd1Zd2rne1e2exrr.r/r&s $zlog._eval_nseries..mul Heaviside),rrrrsympy.core.symbolrr;r2rrmatchr"r#rrr rrrrrrr*r[rrrr7rrrr8rZ nsimplifyr6r!'sympy.functions.special.delta_functionsr, enumeratelseriesras_coeff_exponentrr)$r-rXrrrrrrrrrKr!rrr$rrSrJr_drnZ_resZlogflagsrr&ZptermsrZco1r(rpkrr*r,rlr.rr/rs      "&"           zlog._eval_nseriescCs|jd}tddd}|dkr&d}||||}z|j||dd\}}Wn,tk rz|j|||d} t| YSX||r||||}|dkrt d|t|S|t j kr|t j kr|t j j||dSt||t|} |dkrt|n|}| ||7} |j rt|dkrdd lm} t||D]"\} } | jrZ| d kr@qdq@| d kr| |\}}| d tt| t| d7} | S) NrrTr r0r%rrr+r-r.)r;Ztogetherrrr#rrr2r"r rr8rr6r!r1r,r2r3rr4rr)r-rXrrrrrKcrTrrnr,rlrrr5r.r.r/r*s:        zlog._eval_as_leading_term)r0)r0)N)T)T)r)Nr) r`rarbrqtTupler__annotations__rrrcrdr|r5rrr<rrrrrrrLrrprrGrrNrrrr.r.r.r/r2[s0 !    }  9    ur2cs|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 ========== .. 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