B ˜‘[Pã@s¤dZddlmZddlmZddlmZddlmZm Z dd„Z ej dfd d „Z ej dfd d „Z ej dfd d„Zej dfdd„Zej dfdd„Zej dfdd„ZdS)af Singularities ============= This module implements algorithms for finding singularities for a function and identifying types of functions. The differential calculus methods in this module include methods to identify the following function types in the given ``Interval``: - Increasing - Strictly Increasing - Decreasing - Strictly Decreasing - Monotonic é)Úsympify)Úsolveset)Úsimplify)ÚSÚSymbolcCs>| |¡stdƒ‚n&|jr tjntj}ttd|ƒ||ƒSdS)a& Find singularities of a given function. Currently supported functions are: - univariate rational (real or complex) functions Examples ======== >>> from sympy.calculus.singularities import singularities >>> from sympy import Symbol >>> x = Symbol('x', real=True) >>> y = Symbol('y', real=False) >>> singularities(x**2 + x + 1, x) EmptySet() >>> singularities(1/(x + 1), x) {-1} >>> singularities(1/(y**2 + 1), y) {-I, I} >>> singularities(1/(y**3 + 1), y) {-1, 1/2 - sqrt(3)*I/2, 1/2 + sqrt(3)*I/2} Notes ===== This function does not find nonisolated singularities nor does it find branch points of the expression. References ========== .. [1] http://en.wikipedia.org/wiki/Mathematical_singularity zTAlgorithms finding singularities for non-rational functions are not yet implemented.éN)Zis_rational_functionÚNotImplementedErrorZis_realrÚRealsZ Complexesrr)Ú expressionÚsymbolZdomain©r ú;lib/python3.7/site-packages/sympy/calculus/singularities.pyÚ singularitiess # rNcCsht|ƒ}|j}|dkr*t|ƒdkr*tdƒ‚|p@|r:| ¡ntdƒ}| |¡}t||ƒ|tj ƒ}|  |¡S)zë Helper function for functions checking function monotonicity. It returns a boolean indicating whether the interval in which the function's derivative satisfies given predicate is a superset of the given interval. NrzKThe function has not yet been implemented for all multivariate expressions.Úx) rÚ free_symbolsÚlenrÚpoprÚdiffrrr Z is_subset)r Z predicateÚintervalr ÚfreerZ derivativeZpredicate_intervalr r r Úmonotonicity_helperJs  rcCst|dd„||ƒS)a Return whether the function is increasing in the given interval. Examples ======== >>> from sympy import is_increasing >>> from sympy.abc import x, y >>> from sympy import S, Interval, oo >>> is_increasing(x**3 - 3*x**2 + 4*x, S.Reals) True >>> is_increasing(-x**2, Interval(-oo, 0)) True >>> is_increasing(-x**2, Interval(0, oo)) False >>> is_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval(-2, 3)) False >>> is_increasing(x**2 + y, Interval(1, 2), x) True cSs|dkS)Nrr )rr r r Úxszis_increasing..)r)r rr r r r Ú is_increasingbsrcCst|dd„||ƒS)a‹ Return whether the function is strictly increasing in the given interval. Examples ======== >>> from sympy import is_strictly_increasing >>> from sympy.abc import x, y >>> from sympy import Interval, oo >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.Ropen(-oo, -2)) True >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.Lopen(3, oo)) True >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.open(-2, 3)) False >>> is_strictly_increasing(-x**2, Interval(0, oo)) False >>> is_strictly_increasing(-x**2 + y, Interval(-oo, 0), x) False cSs|dkS)Nrr )rr r r r‘sz(is_strictly_increasing..)r)r rr r r r Úis_strictly_increasing{srcCst|dd„||ƒS)a. Return whether the function is decreasing in the given interval. Examples ======== >>> from sympy import is_decreasing >>> from sympy.abc import x, y >>> from sympy import S, Interval, oo >>> is_decreasing(1/(x**2 - 3*x), Interval.open(1.5, 3)) True >>> is_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo)) True >>> is_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2)) False >>> is_decreasing(-x**2, Interval(-oo, 0)) False >>> is_decreasing(-x**2 + y, Interval(-oo, 0), x) False cSs|dkS)Nrr )rr r r rªszis_decreasing..)r)r rr r r r Ú is_decreasing”srcCst|dd„||ƒS)a Return whether the function is strictly decreasing in the given interval. Examples ======== >>> from sympy import is_strictly_decreasing >>> from sympy.abc import x, y >>> from sympy import S, Interval, oo >>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo)) True >>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2)) False >>> is_strictly_decreasing(-x**2, Interval(-oo, 0)) False >>> is_strictly_decreasing(-x**2 + y, Interval(-oo, 0), x) False cSs|dkS)Nrr )rr r r rÁsz(is_strictly_decreasing..)r)r rr r r r Úis_strictly_decreasing­srcCsdt|ƒ}|j}|dkr*t|ƒdkr*tdƒ‚|p@|r:| ¡ntdƒ}t| |¡||ƒ}| |¡t j kS)a Return whether the function is monotonic in the given interval. Examples ======== >>> from sympy import is_monotonic >>> from sympy.abc import x, y >>> from sympy import S, Interval, oo >>> is_monotonic(1/(x**2 - 3*x), Interval.open(1.5, 3)) True >>> is_monotonic(1/(x**2 - 3*x), Interval.Lopen(3, oo)) True >>> is_monotonic(x**3 - 3*x**2 + 4*x, S.Reals) True >>> is_monotonic(-x**2, S.Reals) False >>> is_monotonic(x**2 + y + 1, Interval(1, 2), x) True NrzKis_monotonic has not yet been implemented for all multivariate expressions.r) rrrrrrrrÚ intersectionrZEmptySet)r rr rrZturning_pointsr r r Ú is_monotonicÄsr)Ú__doc__Zsympy.core.sympifyrZsympy.solvers.solvesetrZsympy.simplifyrZsympyrrrr rrrrrrr r r r Ús   2