a kh@s`ddlmZddlmZmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZmZmZmZmZdd lmZmZmZmZdd lmZdd lmZm Z ddl!m"Z"m#Z#ddl$m%Z%ddl&m'Z'ddl(m)Z)ddl*m+Z+GdddeZ,e+dZ-e-.e/e/fe,ddl.m0Z0ddl1m2Z2m3Z3ddl4m5Z5m6Z6ddl7m8Z8m9Z9m:Z:dS)) annotations)Callable TYPE_CHECKING)product)_sympify)cacheit)S)Expr)PrecisionExhausted)expand_complexexpand_multinomial expand_mul_mexpand PoleError) fuzzy_bool fuzzy_not fuzzy_andfuzzy_or)global_parameters)is_gtis_lt) NumberKind UndefinedKind)sift)sympy_deprecation_warning)as_int) Dispatchercs eZdZdZdZdZer*eddddZeddd d Z eddd d Z ed dZ e d{ddddddZ d|ddZeddZddZddZddZdd Zd!d"Zd#d$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(dIdJZ)d}dKdLZ*dMdNZ+dOdPZ,dQdRZ-dSdTZ.dUdVZ/dWdXZ0dYdZZ1d[d\Z2d]d^Z3d_d`Z4d~dbdcZ5ddedfZ6dgdhZ7e didjZ8fdkdlZ9dmdnZ:dodpZ;dqdrZdwdxZ?dydzZ@ZAS)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 .. 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