o g @s\ddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z dd lmZdd lmZmZmZmZmZdd lmZmZmZmZdd lmZdd lmZmZddl m!Z!m"Z"ddl#m$Z$ddl%m&Z&ddl'm(Z(ddl)m*Z*Gddde Z+e*dZ,e,-e.e.fe+ddl-m/Z/ddl0m1Z1m2Z2ddl3m4Z4m5Z5ddl6m7Z7m8Z8m9Z9dS)) annotations)Callable)product)_sympify)cacheit)S)Expr)PrecisionExhausted)expand_complexexpand_multinomial expand_mul_mexpand PoleError) fuzzy_bool fuzzy_not fuzzy_andfuzzy_or)global_parameters)is_gtis_lt) NumberKind UndefinedKind)sift)sympy_deprecation_warning)as_int) Dispatchercs eZdZUdZdZdZded<ded<edydd Zdzd d Z e d{ddZ e d{ddZ e ddZ eddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.Zd/d0Zd1d2Zd3d4Zd5d6Zd7d8Z d9d:Z!d;d<Z"d=d>Z#d?d@Z$dAdBZ%dCdDZ&dEdFZ'dGdHZ(d|dIdJZ)dKdLZ*dMdNZ+dOdPZ,dQdRZ-dSdTZ.dUdVZ/dWdXZ0dYdZZ1d[d\Z2d]d^Z3d}d`daZ4d~dcddZ5ddedfZ6edgdhZ7fdidjZ8dkdlZ9dmdnZ:dodpZ;dqdrZdwdxZ?Z@S)Powa% Defines the expression x**y as "x raised to a power y" .. deprecated:: 1.7 Using arguments that aren't subclasses of :class:`~.Expr` in core operators (:class:`~.Mul`, :class:`~.Add`, and :class:`~.Pow`) is deprecated. See :ref:`non-expr-args-deprecated` for details. Singleton definitions involving (0, 1, -1, oo, -oo, I, -I): +--------------+---------+-----------------------------------------------+ | expr | value | reason | +==============+=========+===============================================+ | z**0 | 1 | Although arguments over 0**0 exist, see [2]. | +--------------+---------+-----------------------------------------------+ | z**1 | z | | +--------------+---------+-----------------------------------------------+ | (-oo)**(-1) | 0 | | +--------------+---------+-----------------------------------------------+ | (-1)**-1 | -1 | | +--------------+---------+-----------------------------------------------+ | S.Zero**-1 | zoo | This is not strictly true, as 0**-1 may be | | | | undefined, but is convenient in some contexts | | | | where the base is assumed to be positive. | +--------------+---------+-----------------------------------------------+ | 1**-1 | 1 | | +--------------+---------+-----------------------------------------------+ | oo**-1 | 0 | | +--------------+---------+-----------------------------------------------+ | 0**oo | 0 | Because for all complex numbers z near | | | | 0, z**oo -> 0. | +--------------+---------+-----------------------------------------------+ | 0**-oo | zoo | This is not strictly true, as 0**oo may be | | | | oscillating between positive and negative | | | | values or rotating in the complex plane. | | | | It is convenient, however, when the base | | | | is positive. | +--------------+---------+-----------------------------------------------+ | 1**oo | nan | Because there are various cases where | | 1**-oo | | lim(x(t),t)=1, lim(y(t),t)=oo (or -oo), | | | | but lim( x(t)**y(t), t) != 1. See [3]. | +--------------+---------+-----------------------------------------------+ | b**zoo | nan | Because b**z has no limit as z -> zoo | +--------------+---------+-----------------------------------------------+ | (-1)**oo | nan | Because of oscillations in the limit. | | (-1)**(-oo) | | | +--------------+---------+-----------------------------------------------+ | oo**oo | oo | | +--------------+---------+-----------------------------------------------+ | oo**-oo | 0 | | +--------------+---------+-----------------------------------------------+ | (-oo)**oo | nan | | | (-oo)**-oo | | | +--------------+---------+-----------------------------------------------+ | oo**I | nan | oo**e could probably be best thought of as | | (-oo)**I | | the limit of x**e for real x as x tends to | | | | oo. If e is I, then the limit does not exist | | | | and nan is used to indicate that. | +--------------+---------+-----------------------------------------------+ | oo**(1+I) | zoo | If the real part of e is positive, then the | | (-oo)**(1+I) | | limit of abs(x**e) is oo. So the limit value | | | | is zoo. | +--------------+---------+-----------------------------------------------+ | oo**(-1+I) | 0 | If the real part of e is negative, then the | | -oo**(-1+I) | | limit is 0. | +--------------+---------+-----------------------------------------------+ Because symbolic computations are more flexible than floating point calculations and we prefer to never return an incorrect answer, we choose not to conform to all IEEE 754 conventions. This helps us avoid extra test-case code in the calculation of limits. See Also ======== sympy.core.numbers.Infinity sympy.core.numbers.NegativeInfinity sympy.core.numbers.NaN References ========== .. [1] https://en.wikipedia.org/wiki/Exponentiation .. [2] https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero .. [3] https://en.wikipedia.org/wiki/Indeterminate_forms Tis_commutativeztuple[Expr, Expr]args_argsNcCs:|durtj}t|}t|}ddlm}t||st||r#td||fD]}t|ts=tdt |j ddddd q'|r|t j urIt j S|t jurzt|t jrWt jSt|t jrft|t jrft jSt|t jrz|jrrt j S|jd urzt j S|t jurt jS|t jur|S|d kr|st j S|jj d kr|t jkrd dlm}|t||jt||jSn'|jr|js|jr|jr|j s|j!r|"r|j#r| }n |j$rt| | St j ||fvrt j S|t jurt%|j&rt j St jSd dl'm(}|j)ss|t jurst||ssddl*m+}d dl'm,} d dl-m.} ||d d/\} } | | \} }t|| r?|j0d |kr?t j| | S|j1rsd dl2m3}m4}|||}|j!rs|rs|| ||d d |t j5t j6krst j| | S|7|}|dur|St8|||}|9|}t|ts|S|j:o|j:|_:|S)Nr) Relationalz Relational cannot be used in Powzf Using non-Expr arguments in Pow is deprecated (in this case, one of the arguments is of type zf). If you really did intend to construct a power with this base, use the ** operator instead.z1.7znon-expr-args-deprecated)deprecated_since_versionactive_deprecations_target stacklevelFAccumulationBoundsr AccumBounds) exp_polar) factor_termslog)fraction)sign)r0im);revaluater relationalr" isinstance TypeErrorr rtype__name__rComplexInfinityNaNInfinityrOne NegativeOnerZero is_finite __class__Exp1!sympy.calculus.accumulationboundsr*rminmax is_Symbol is_integer is_Integer is_numberis_Mul is_Numbercould_extract_minus_signis_evenis_oddabs is_infinite&sympy.functions.elementary.exponentialr+is_Atom exprtoolsr,r.sympy.simplify.radsimpr/ as_coeff_Mulr is_Add$sympy.functions.elementary.complexesr0r1 ImaginaryUnitPi _eval_power__new__ _exec_constructor_postprocessorsr)clsber2r"argr*r+r,r.r/cexnumdenr0r1sobjreG/var/www/visachat/venv/lib/python3.10/site-packages/sympy/core/power.pyrYvs                             z Pow.__new__rcCs |jtjkrddlm}|SdSNrr-)baserr@rOr.)selfargindexr.rererfinverses  z Pow.inversereturnr cC |jdSNrr!rirererfrh zPow.basecCrmNrrorprererfexprqzPow.expcCs|jjtur |jjStSN)rskindrrhrrprererfrus zPow.kindcCs dd|jfS)N)r7r[rererf class_keys z Pow.class_keycCszddlm}m}|\}}||||r7|r9||||r(t| |S||||r;t| | SdSdSdS)Nr)askQ) sympy.assumptions.askrzr{ as_base_expintegerrJevenrodd)ri assumptionsrzr{r\r]rererf _eval_refines  zPow._eval_refinecCsj|\}}|tjur|||Sd}|jrd}n |jr!d}n|jdur%ddlm}m}m }m }ddl m } m } ddlm} dd} dd } |jr|d krr| |rq|jd urftj|t| ||S|jd urqt|| Sn|jr|jr|t|}|jrt||tj}t|dkd ks|dkrd}n|jrd}n||jrt|d kd krd}nx| |r| d tjtj|| tj|||d tj}|jr| |||dkr||}nFd}nCz6| d tjtj|| tj||| |d tj}|jr| |||dkr||}nd}Wn ty$d}Ynw|dur3|t|||SdS)Nrr)r^r1rer0rsr.)floorcSs:t|dddkr dS|\}}|jr|dkrdSdSdS)zZReturn True if the exponent has a literal 2 as the denominator, else None.qNrwT)getattras_numer_denomrE)r]ndrererf_half s  zPow._eval_power.._halfcSs6z|jddd}|jr|WSWdStyYdSw)zXReturn ``e`` evaluated to a Number with 2 significant digits, else None.rwTstrictN)evalfrIr )r]rvrererf_n2s zPow._eval_power.._n2r'TFrw)r}rr9rEis_polaris_extended_realrUr^r1rr0rOrsr.#sympy.functions.elementary.integersr is_negativer<rrKrM is_imaginaryrVis_extended_nonnegativerWHalfr )riotherr\r]rcr^r1rr0rsr.rrrrererfrXsn          "  zPow._eval_powerc Cst|j|j}}|jr|jr|jr||dkrtjSddlm}|jrd|jrd|jrdt |t |t |}}}| }|dkr\||kr\| d|kr\t ||} t t || || |St t |||Sddl m} t|tr|jr|jr| ||}| t||dd|St|tr|jr|jrt | } | dkr||} | | || }| t||dd|Sd Sd Sd Sd Sd Sd S) aOA dispatched function to compute `b^e \bmod q`, dispatched by ``Mod``. Notes ===== Algorithms: 1. For unevaluated integer power, use built-in ``pow`` function with 3 arguments, if powers are not too large wrt base. 2. For very large powers, use totient reduction if $e \ge \log(m)$. Bound on m, is for safe factorization memory wise i.e. $m^{1/4}$. For pollard-rho to be faster than built-in pow $\log(e) > m^{1/4}$ check is added. 3. For any unevaluated power found in `b` or `e`, the step 2 will be recursed down to the base and the exponent such that the $b \bmod q$ becomes the new base and $\phi(q) + e \bmod \phi(q)$ becomes the new exponent, and then the computation for the reduced expression can be done. r)totientPr#r)ModFr2N)rhrsrE is_positiverr=%sympy.functions.combinatorial.numbersrrFint bit_lengthIntegerpowmodrr4rrG) rirrhrsrr\r]mmbphirrrererf _eval_ModOs2        z Pow._eval_ModcCs |jjr |jjr|jjSdSdSrt)rsrErrhrKrprererf _eval_is_evenszPow._eval_is_evencCst|}|dur |jS|SNT)r_eval_is_extended_negativer>)riext_negrererf_eval_is_negatives zPow._eval_is_negativecCs|j|jkr|jjr dSdS|jjr|jjrdSdS|jjr,|jjr$dS|jjr*dSdS|jjr:|jj r8|jjSdS|jj rF|jjrDdSdS|jj rr|jj rb|jd}|jrXdS|j rb|jdurbdS|jj rtddl m}||jj SdSdS)NTFr#rr-)rhrsrris_realis_extended_negativerKrLis_zeroris_extended_nonpositiverrErOr.)rirr.rererf_eval_is_extended_positivesD    zPow._eval_is_extended_positivecCs|jtjur|jjs|jjrdS|jjr&|jjr|jjrdS|jj r$dSdS|jj r2|jjr0dSdS|jj r>|jjrrKis_extended_positiverrrrprererfrs< zPow._eval_is_extended_negativecCs|jjr|jjr dS|jjrdSdS|jtjkr|jtjuS|jjdurb|jjr.|jjr.dS|jj r6|jj S|jj rrrNis_nonnegativerrMrrprererf _eval_is_zeros2   zPow._eval_is_zerocCs|j\}}|jr|jdur|jrdS|jr'|jr'|tjurdS|js%|jr'dS|jrC|jrC|js3|jrCt |dj rCt |dj rCdS|j rR|j rR|j |j}|j S|jr_|jr_|djr_dS|jrl|jrn|djrpdSdSdSdS)NFTr)r is_rationalrErrr<rrr>rrrIfuncrF)rir\r]checkrererf_eval_is_integers(       zPow._eval_is_integerc Cs|jtjur|jjr dS|jjrdtj|jtjjSddl m }m}|jj}|durS|jj |kr;|jjjr;|jjS|jj t krQ|jjtjurQ|jjjrQ|jjSdS|jj}|dur]dS|r|r|jj rgdS|jjrq|jjrqdS|jjr{|jjr{dS|jjr|jjrdS|jjr|jjrdS|r|jjr|jjdurt |j|j jS|jj}|jj}|r|jjr|jjrdS|jjrdSn=|r||jjrdS|jjr|j\}}|r|jrt|j||j|ddjSn|jtj tjfvr|jdjdurdS|rB|rB|jtjur dS|jtj}|rB|jjr0|jr0|jjr0|jdjr0|jr0dS|||jtjj} | durB| S|durr|rtt|jtrZ|jj dkrZdSddl!m"} | |j|jtj} | j#rv| jSdSdSdS) NTrwr)r.rsFrrr^)$rhrr@rsrrrVrWrKrOr.rrrrrEis_extended_nonzerorr is_RationalrrLrT as_coeff_AddrFMulr<coeffr is_nonzeror4RationalprUr^r) rir.rsreal_breal_eim_bim_er_aokr^irererf_eval_is_extended_reals $     zPow._eval_is_extended_realcCsH|jtjkrt|jj|jjgStdd|jDr | r"dSdSdS)Ncs|]}|jVqdSrt)r).0rrererf ?z'Pow._eval_is_complex..T) rhrr@rrsrrallr _eval_is_finiterprererf_eval_is_complex:s zPow._eval_is_complexc Cs>|jjdurdS|jjr|jjr|jj}|dur|SdS|jtjkr9d|jtjtj }|j r2dS|jr7dSdS|jjrOddl m }||jj}|durOdS|jj ry|jj ry|jjr]dS|jj}|se|S|jjrkdSd|jj}|rw|jjS|S|jj durddlm}||j|jtj}d|j} | dur| SdSdS)NFrwTrr-r)rhrrrsrErLrr@rWrVrKrOr.rrrrrUr^) rirfr.imlograthalfr^risoddrererf_eval_is_imaginaryBsP        zPow._eval_is_imaginarycCsD|jjr|jjr |jjS|jjr|jjrdS|jtjur dSdSdSr)rsrErrhrLrrr<rprererf _eval_is_oddss zPow._eval_is_oddcCs||jjr|jjr dS|jjs|jjrdS|jj}|durdS|jj}|dur(dS|r8|r:|jjs6t|jjrrr)ric1c2rererfr|s zPow._eval_is_finitecCs,|jjr|jjr|jdjrdSdSdSdS)zM An integer raised to the n(>=2)-th power cannot be a prime. rFN)rhrErsrrprererf_eval_is_primes zPow._eval_is_primecCs\|jjr$|jjr&|jdjr|jdjs"|jdjr(|jjr*|jjr,dSdSdSdSdSdS)zS A power is composite if both base and exponent are greater than 1 rTN)rhrErsrrrKrprererf_eval_is_composites   zPow._eval_is_compositecCs|jjSrt)rhrrprererf_eval_is_polarszPow._eval_is_polarcCsddlm}t|j|r*|j||}|j||}t||r$||S|||Sddlm}m }dd}||jksE||kr_|jt j kr_|j rVt|t rV||j||S||j||St||jrz|j|jkrz||j|j} | jrzt|| St||jr#|j|jkr#|jjdur|jjtdd} |jjtdd} || | |\} } }| r||| }|durt|t|j|}|Snd|j}g}g}|} |jjD]6}|||}|} || | |\} } }| r||| |dur||q|js|jsdS||q|r#t|}||dkrt|j|dd n|jt|St||s4|jrw|jt j ury|jjru|jjr{|jjtdd} |j||jjtdd} || | |\} } }| r}||| }|durst|t|j|}|SdSdSdSdSdS) Nrr)rc Ss |\}}|\}}||kr|jr>||}z t|ddd}Wnty8|\} } | jr0| jp5| jo5| j}Ynw||dfSt|tsF|f}t dd|DsQdSz2t t|t|\}} |dkro| dkro|d 7}| t|8} | dkrvd} nt | g|R} d|| fWStyYdSwdS) a*Return (bool, pow, remainder_pow) where, if bool is True, then the exponent of Pow `old` will combine with `pow` so the substitution is valid, otherwise bool will be False. For noncommutative objects, `pow` will be an integer, and a factor `Pow(old.base, remainder_pow)` needs to be included. If there is no such factor, None is returned. For commutative objects, remainder_pow is always None. cti are the coefficient and terms of an exponent of self or old In this _eval_subs routine a change like (b**(2*x)).subs(b**x, y) will give y**2 since (b**x)**2 == b**(2*x); if that equality does not hold then the substitution should not occur so `bool` will be False. FrTNcsrrt)rE)rtermrererfrrz1Pow._eval_subs.._check..)FNNrr) rr ValueErrorr}rrrr4tuplerdivmodr) ct1ct2oldcoeff1terms1coeff2terms2rcombinesr\r] remainder remainder_powrererf_checks>        zPow._eval_subs.._checkF)as_Addrr)rAr*r4rsrhsubs__rpow__rrOr.rr@ is_Functionr_subsrIrrTas_independentSymbolr as_coeff_mulr appendrrEAddis_Powrr)rirnewr*r\r]rsr.rlrrrrrresultoargnew_lo_alrnewaexporererf _eval_subssz     :        &6  zPow._eval_subscCs<|j\}}|jr|j|jkr|jdkrd|| fS||fS)aReturn base and exp of self. Explanation =========== If base a Rational less than 1, then return 1/Rational, -exp. If this extra processing is not needed, the base and exp properties will give the raw arguments. Examples ======== >>> from sympy import Pow, S >>> p = Pow(S.Half, 2, evaluate=False) >>> p.as_base_exp() (2, -2) >>> p.args (1/2, 2) >>> p.base, p.exp (1/2, 2) rr)r rrr)rir\r]rererfr} s zPow.as_base_expcCszddlm}|jj|jj}}|r||j|jS|r#|j||jS|dur7|dur9t|}||kr;||SdSdSdS)Nr)adjointF)rUrrsrErhrr )rirrrexpandedrererf _eval_adjoint=s zPow._eval_adjointcCs|ddlm}|jj|jj}}|r||j|jS|r#|j||jS|dur7|dur7t|}||kr7||S|jr<|SdS)Nr) conjugateF)rUrrsrErhrr r)rir_rrrrererf_eval_conjugateIs zPow._eval_conjugatecCsddlm}|jtjkr|tj|jS|jj|jjp |jj }}|r+|j|jS|r5||j|jS|durI|durKt |}||krM||SdSdSdS)Nr) transposeF) rUrrhrr@rrsrErrNr )rirrrrrererf_eval_transposeWs   zPow._eval_transposec sjj}tjkr-ddlm}t||r-|jr-ddlm }| |j g|j RS|j rs|dds?jdus?|rs|jrOtfdd|jDSjrst|jdd d d \}}|rstfd d|Dt|SS) za**(n + m) -> a**n*a**mr)Sum)ProductforceFcg|]}|qSrerrxr\rirerf qz.Pow._eval_expand_power_exp..cSs|jSrtrrrererfssz,Pow._eval_expand_power_exp..Tbinarycr rerrrrerfrur)rhrsrr@sympy.concrete.summationsr r4rsympy.concrete.productsr 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