\c@ sddlmZmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZdd lmZmZmZmZdd lmZmZdd lmZmZdd lmZdd lmZddlm Z!ddlm"Z#dZ$dZ%dZ&de fdYZ'ddl(m)Z)ddl*m+Z+ddl,m-Z-m.Z.ddl/m0Z0m1Z1m2Z2dS(i(tprint_functiontdivision(tlogi(t_sympify(tcacheit(tS(tExpr(tPrecisionExhausted(t _coeff_isnegtexpand_complextexpand_multinomialt expand_mul(t fuzzy_boolt fuzzy_not(tas_inttrange(tglobal_evaluate(tsift(tsqrtrem(tsqrtcC s3|dkrtt|Stt|ddS(s9Return the largest integer less than or equal to sqrt(n).l,WV8ii(tintt_sqrttinteger_nthroot(tn((s/lib/python2.7/site-packages/sympy/core/power.pytisqrts c C s.t|t|}}|dkr4tdn|dkrOtdn|d kre|tfS|dkr{|tfS|dkrt|\}}t|| fS||krdtfSyt|d|d}Wnotk rMt|d|}|dkr:t|d}td ||d|>}qNtd |}nX|d krd |}}xV||d}||d||||}}t||dkrjPqjqjWn|}||}x$||kr|d7}||}qWx$||kr|d8}||}qWt|||kfS(s@ Return a tuple containing x = floor(y**(1/n)) and a boolean indicating whether the result is exact (that is, whether x**n == y). Examples ======== >>> from sympy import integer_nthroot >>> integer_nthroot(16, 2) (4, True) >>> integer_nthroot(26, 2) (5, False) To simply determine if a number is a perfect square, the is_square function should be used: >>> from sympy.ntheory.primetest import is_square >>> is_square(26) False See Also ======== sympy.ntheory.primetest.is_square integer_log isy must be nonnegativeisn must be positiveig?g?i5g@i2i(iiI( Rt ValueErrortTruetmpmath_sqrtremRtFalset OverflowErrort_logtabs( tyRtxtremtguesstexptshifttxprevtt((s/lib/python2.7/site-packages/sympy/core/power.pyRsL             !    c C s|dkrtdn|dkr6tdn|dkr~t|}t|}|jd}||||kfS|dkrt|dkr|n| | \}}||ot|dkr|dn|d fSt|}t|}d}}x}||kr|}d}x^||krt||\}}|pN|}||7}||kr$||9}|d9}q$q$Wq W||dko|dkfS(s Returns (e, bool) where e is the largest nonnegative integer such that |y| >= |x**e| and bool is True if y == x**e Examples ======== >>> from sympy import integer_log >>> integer_log(125, 5) (3, True) >>> integer_log(17, 9) (1, False) >>> integer_log(4, -2) (2, True) >>> integer_log(-125,-5) (3, True) See Also ======== integer_nthroot sympy.ntheory.primetest.is_square sympy.ntheory.factor_.multiplicity sympy.ntheory.factor_.perfect_power isx cannot take value as 1isy cannot take value as 0ii(ii(RRRt bit_lengtht integer_logtbooltdivmod( R R!teRtbtrtdtmR"((s/lib/python2.7/site-packages/sympy/core/power.pyR)gs4      )1       tPowcB seZdZeZdgZed0dZe dZ e dZ e dZ dZdZdZd Zd Zd Zd Zd ZdZdZdZdZdZdZdZdZdZdZdZ dZ!dZ"dZ#dZ$dZ%edZ&dZ'd Z(d!Z)d"Z*d#Z+d$Z,d%Z-d&Z.d'Z/ie0d(Z1d)Z2d*Z3ed+Z4d,Z5e0ed-Z6d.Z7d/Z8RS(1sD Defines the expression x**y as "x raised to a power y" 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 that 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/Exponentiation#Zero_to_the_power_of_zero .. [3] https://en.wikipedia.org/wiki/Indeterminate_forms tis_commutativecC s|dkrtd}nddlm}t|}t|}|r|tjkr]tjS|tjkrstj S|tj kr|S|j s|j r|j s|j r|j rt |r|jr| }q|jrt| | Sntj||fkr tjS|tj kr5t|jr.tjStj S|j rc|tjk rct|| rcddlm}m}m}m}m} m} | |dtj\} } || } t| |r| jd|krtj| || S| j rc|| |}|j!r`|r`| || |dt |tj"tj#kr`tj| || Sqcn|j$|}|dk r|Snt%j&|||}|j'|}t|ts|S|j(o|j(|_(|S(Nii(t exp_polar(tnumertdenomRtsigntimt factor_termsR6()tNoneRt&sympy.functions.elementary.exponentialR3RRtComplexInfinitytNaNtZerotOnet is_Symbolt is_numbert is_integerRtis_eventis_oddR1Rt is_infinitetis_AtomtExp1t isinstancetsympyR4R5RR6R7R8Rt as_coeff_Multargstis_Addt is_Numbert ImaginaryUnittPit _eval_powerRt__new__t _exec_constructor_postprocessorsR2(tclsR-R,tevaluateR3R4R5RR6R7R8tctextdentstobj((s/lib/python2.7/site-packages/sympy/core/power.pyRPsX    $   ). 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jr||j| |j| dtjSq|jtj tjfkr|jdjtkrtSqn|r|r|jtjkr=t S|jjtj} | r| ||jtjj} | dk r| Sqn|tkr||j|jtj} | jSdS(Ni(RhR$RtMuliRSi(RHRhR$RRR[RtR9RRJRxR~RRyRARut is_RationalRR1RBRCRKt as_coeff_AddRRRMRvtcoeffRN( RZRhR$RRtreal_btreal_etim_btim_eRTtatokti((s/lib/python2.7/site-packages/sympy/core/power.pyt _eval_is_realsf"  %                 $     cC s!td|jDrtSdS(Ncs s|]}|jVqdS(N(t is_complex(t.0R((s/lib/python2.7/site-packages/sympy/core/power.pys 9s(tallRJR(RZ((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_is_complex8sc C saddlm}m}|jjrQ|jjrQ|jj}|dk rJ|SdSn|jjr||jj}|dk rt Sn|jj r|jj r|jj rt S|jj }|s|S|jjrt Sd|jj}|r|jj S|Sn|jj t krF||j|jtj}d|j}|dk rF|Sn|jj r]d|jSdS(Ni(RhRii(RHRhRR[RxR$RARCR9RRtR~RRuRRN( RZRhRRdtimlogtratthalfRtisodd((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_is_imaginary<s<             cC s[|jjrW|jjr"|jjS|jjr>|jjr>tS|jtjkrWtSndS(N( R$RAR~R[RCRyRRRv(RZ((s/lib/python2.7/site-packages/sympy/core/power.pyt _eval_is_oddbs   cC s|jjr/|jjrtS|jjr/tSn|jj}|dkrKdS|jj}|dkrgdS|r|r|jj st |jjrtSndS(N( R$RuR[RzRRDRRR9RyR (RZtc1tc2((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_is_finiteks        cC s0|jjr,|jjr,|jdjr,tSdS(sM An integer raised to the n(>=2)-th power cannot be a prime. iN(R[RAR$R~R(RZ((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_is_prime{s(cC sh|jjrd|jjrd|jdjr8|jdjs`|jdjrd|jjrd|jjrdtSdS(sS A power is composite if both base and exponent are greater than 1 iN(R[RAR$R~RuRBR(RZ((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_is_composites (cC s |jjS(N(R[Rs(RZ((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_is_polarsc sbddlmm}mfd}||jkrT||jj||St||jr|j|jkr||j|j}|jrt ||Snt||jr|j|jkr|jj t krp|jj dt }|jj dt }||||\}} } |r|j|| } | dk rit| t |j| } n| Sq|j} g} g}| j}x|jjD]}|j||}|j}||||\}} } |r| j|| | dk r|j| qqn|j r.|j r.dS|j|qW| rt|}| j|dkr{t |j|dt n|jt| Snt|r^|jjr^|jjr^|jdj dt }|j||jj dt }||||\}} } |r^|j|| } | dk rWt| t |j| } n| SndS(Ni(R$RtSymbolc  s|\}}|\}}||kr|jr||}yt|dt}t}WnGtk rttjtdt|j|tf}nX||dfSt|t s|f}nt d|DstddfSyt t|t|\}} |dkrA| dkrA|d7}| t|8} n| dkrVd} nt | |} t|| fSWqtk rqXntddfS(s*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. RoRScs s|]}|jVqdS(N(RA(Rtterm((s/lib/python2.7/site-packages/sympy/core/power.pys siiN(R2RRRRRGR1RORaR9ttupleRR+R( tct1tct2toldtcoeff1tterms1tcoeff2tterms2Rtcombinest remaindert remainder_pow(RR$(s/lib/python2.7/site-packages/sympy/core/power.pyt_checks:           !    tas_AddiRSi(RHR$RRR[t_subsRGRRLR1RKRtas_independentR9Rt as_coeff_mulRJtappendR2RAtAddRtR~(RZRtnewRRtlRRRRRtresulttoargtnew_lto_alRtnewatexpo((RR$s/lib/python2.7/site-packages/sympy/core/power.pyt _eval_subss`9$ $      4 ' cC sT|j\}}|jrJ|jdkrJ|jdkrJt|j| fS||fS(sReturn base and exp of self. If base is 1/Integer, then return Integer, -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) i(RJRtpRktInteger(RZR-R,((s/lib/python2.7/site-packages/sympy/core/power.pyRas'cC sddlm}|jj|jj}}|rC||j|jS|r]|j||jS|tkr|tkrt|}||kr||SndS(Ni(tadjoint(t$sympy.functions.elementary.complexesRR$RAR[R~RR (RZRRRtexpanded((s/lib/python2.7/site-packages/sympy/core/power.pyt _eval_adjoints  cC sddlm}|jj|jj}}|rC||j|jS|r]|j||jS|tkr|tkrt|}||kr||Sn|jr|SdS(Ni(Rw( RRwR$RAR[R~RR Rt(RZRTRRR((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_conjugate%s    cC sddlm}|jj|jj}}|r=|j|jS|rW||j|jS|tkr|tkrt|}||kr||SndS(Ni(t transpose(RRR$RAR[RRR (RZRRRR((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_transpose3s  cK st|j}|j}|jrd|jrdg}x-|jD]"}|j|j|j|q4Wt|S|j||S(sa**(n + m) -> a**n*a**m(R[R$RKR2RJRRR(RZthintsR-R,texprR!((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_expand_power_exp?s    c K s]|jdt}|j}|j}|js1|S|jdt\}}|r5g|D]*}t|drz|j|n|^qV}|jr|j rt ||}n5t g|dddD]}|d^q| }|r|t ||9}n|S|s#|j t ||dtSt |g}nt |ddt \} } d } t | | } | t } | | d7} | t}| tj}|rtj}t|d }|d krn|d kr| j|n|d kr8|r%|j }|tjk r5| j|q5q|jtjnR|rm|j }|tjk r}| j|q}n|jtj| j|~n|s|jr| || }|} n1|j stt|d krftj}| r |d jr ||jd 9}nt|d r%| }nx|D]}| j| q,W|tjk r| j|qnn|r| r|d jr|d tjk r| jtj| j|d  q| j|n | j|~| }| |7} tj}|r.|t g|D]}|j ||dt^q9}n| rY||j t | |dt9}n|S(s(a*b)**n -> a**n * b**ntforcetsplit_1t_eval_expand_power_baseNiRScS s |jtkS(N(RtR(R!((s/lib/python2.7/site-packages/sympy/core/power.pytlttbinarycS sF|tjkrtjS|j}|r)tS|dkrBt|jSdS(N(RRMRsRR9R Ry(R!tpolar((s/lib/python2.7/site-packages/sympy/core/power.pytpredns  iiii(tgetRR[R$tis_Multargs_cncthasattrRRR~RRRRR9RRMtlenRtpopR>RvRAtAssertionErrorRLtextend(RZRRR-R,tcargstncRRqR|t maybe_realRtsiftedtnonnegtnegtimagtItnonntoR((s/lib/python2.7/site-packages/sympy/core/power.pyRJs   4  5                    8%cK sa|j\}}|}|jr|jdkr|jr|jst|j|j}|s_|S|j|||g}}|j||}|jr|j }nx(t j |D]}|j ||qWt |Snt |}|jrgg} } x7|jD],} | jr)| j | q | j | q W| rt | } t | } |dkrt| |dt|| | St| |ddt}t| |dt||| Sn|jr|j\}} |jr| jr|jso| js@|j|j| j|}|j| j|j| j}} q|j|j|}|j|j| }} n>| js|j| j|}|| j| j}} nd}t |t | ddf\}} }}xq|rJ|d@r||| || |||}}|d8}n||| | d|| }} |d}qWtj}|dkrl|||St|||||Sqn| }ddlm}ddlm}|t||}|||S|dkrt g|jD] } |jD]}| |^qqS||dj }|jret g|jD] } |jD]}| |^qKq>St g|jD]} | |^qrSn|jr|jdkr|jrt|j|jkrd|j|| j S|jrY|jrYtjtj}}x=|jD]2}|jr7||j||9}q||7}qW||j||S|SdS( sA(a + b + ..)**n -> a**n + n*a**(n-1)*b + .., n is nonzero integeriitdeepii(tmultinomial_coefficients(tbasic_from_dictN( RJRRRKRRRkRtis_Powt_eval_expand_multinomialRt make_argsRRR2tis_OrderR RR R@t as_real_imagRRMRHRtsympy.polys.polyutilsRRRRLR>R=(RZRR[R$RRtradicaltexpanded_base_nRt order_termst other_termsR-tfRtgRtkRTR/RRRRtexpansion_dicttmultiRttail((s/lib/python2.7/site-packages/sympy/core/power.pyRs! 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         ,      )  $         +    (         cC sjddlm}m}|jj|sG|j|jj||jS||j||jj|S(Ni(R$R(RHR$RRRR[R>(RZR!R$R((s/lib/python2.7/site-packages/sympy/core/power.pyt_eval_as_leading_term scG s0ddlm}||j||j||S(Ni(tbinomial(RHR]R$R(RZRR!tprevious_termsR]((s/lib/python2.7/site-packages/sympy/core/power.pyRGscC s"|jdj|jdjS(Nii(RJt_sage_(RZ((s/lib/python2.7/site-packages/sympy/core/power.pyR_scC s|j\}}t|jd|d|}|jd|d|\}}|jr|j\}}|jr||} |j|| } tj} | jst| j | j \} } |j|| } n| |j|t||| || j fSnt||}|jr|j r|jd|d|\}}|j||j \} } | j\} }| tj ks||kr| |jt| ||fSntj |j||fS(sReturn the tuple (R, self/R) where R is the positive Rational extracted from self. Examples ======== >>> from sympy import sqrt >>> sqrt(4 + 4*sqrt(2)).as_content_primitive() (2, sqrt(1 + sqrt(2))) >>> sqrt(3 + 3*sqrt(2)).as_content_primitive() (1, sqrt(3)*sqrt(1 + sqrt(2))) >>> from sympy import expand_power_base, powsimp, Mul >>> from sympy.abc import x, y >>> ((2*x + 2)**2).as_content_primitive() (4, (x + 1)**2) >>> (4**((1 + y)/2)).as_content_primitive() (2, 4**(y/2)) >>> (3**((1 + y)/2)).as_content_primitive() (1, 3**((y + 1)/2)) >>> (3**((5 + y)/2)).as_content_primitive() (9, 3**((y + 1)/2)) >>> eq = 3**(2 + 2*x) >>> powsimp(eq) == eq True >>> eq.as_content_primitive() (9, 3**(2*x)) >>> powsimp(Mul(*_)) 3**(2*x + 2) >>> eq = (2 + 2*x)**y >>> s = expand_power_base(eq); s.is_Mul, s (False, (2*x + 2)**y) >>> eq.as_content_primitive() (1, (2*(x + 1))**y) >>> s = expand_power_base(_[1]); s.is_Mul, s (True, 2**y*(x + 1)**y) See docstring of Expr.as_content_primitive for more examples. 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