U Zʹj @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) DispatchercseZdZUdZdZdZded<ded<edydd Zdzd d Z e d dddZ e d dddZ 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}dcddZ5d~dedfZ6edgdhZ7fdidjZ8dkdlZ9dmdnZ: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 .. 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Fr}TNcss|] }|jVqdSrh)rB)rtermr]r]r^rsz1Pow._eval_subs.._check..)FNNrr) rr ValueErrorrprrrr1tuplerdivmodr) ct1ct2oldZcoeff1Zterms1Zcoeff2Zterms2rZcombinesrUrV remainder remainder_powr]r]r^_checks8       zPow._eval_subs.._checkF)Zas_Addrr)r>r'r1rgr`subs__rpow__rrLr+rr=Z is_Functionr_subsrFrrNZas_independentSymbolrZ as_coeff_mulr appendrrBAddis_Powrr)rarnewr'rUrVrgr+rlrrrrrresultZoargZnew_lZo_alrZnewaexpor]r]r^ _eval_subssv     :       &6  zPow._eval_subscCs<|j\}}|jr4|j|jkr4|jdkr4d|| 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. 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