o à8Vai,ã@s¦dZddlmZmZddlmZddlmZddlm Z ddd„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)ÚSÚSymbol)Úsympify)Úsolveset)Ú filldedentNc Csâddlm}ddlm}m}m}m}m}ddlm } |dur'|j r$t j nt j }z>> from sympy.calculus.singularities import singularities >>> from sympy import Symbol, log >>> 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} >>> singularities(log(x), x) {0} r)Úlog)ÚsecÚcscÚcotÚtanÚcos)ÚPowNzl Methods for determining the singularities of this function have not been developed.)Z&sympy.functions.elementary.exponentialrZ(sympy.functions.elementary.trigonometricrr r r r Zsympy.core.powerr Zis_realrÚRealsZ ComplexesÚEmptySetZrewriteZatomsZexpZ is_infiniteÚNotImplementedErrorZ is_negativerÚbaseÚargsr) Ú expressionÚsymbolÚdomainrrr r r r r ZsingsÚi©rú>/usr/lib/python3/dist-packages/sympy/calculus/singularities.pyÚ singularitiess& = €  ÿrcCsht|ƒ}|j}|durt|ƒdkrtdƒ‚|p |r| ¡ntdƒ}| |¡}t||ƒ|tj ƒ}|  |¡S)aà Helper function for functions checking function monotonicity. Parameters ========== expression : Expr The target function which is being checked predicate : function The property being tested for. The function takes in an integer and returns a boolean. The integer input is the derivative and the boolean result should be true if the property is being held, and false otherwise. interval : Set, optional The range of values in which we are testing, defaults to all reals. symbol : Symbol, optional The symbol present in expression which gets varied over the given range. It returns a boolean indicating whether the interval in which the function's derivative satisfies given predicate is a superset of the given interval. Returns ======= Boolean True if ``predicate`` is true for all the derivatives when ``symbol`` is varied in ``range``, False otherwise. NézKThe function has not yet been implemented for all multivariate expressions.Úx) rÚ free_symbolsÚlenrÚpoprÚdiffrrrZ is_subset)rZ predicateÚintervalrÚfreeÚvariableZ derivativeZpredicate_intervalrrrÚmonotonicity_helperps ÿ  r#cCót|dd„||ƒS)a Return whether the function is increasing in the given interval. Parameters ========== expression : Expr The target function which is being checked. interval : Set, optional The range of values in which we are testing (defaults to set of all real numbers). symbol : Symbol, optional The symbol present in expression which gets varied over the given range. Returns ======= Boolean True if ``expression`` is increasing (either strictly increasing or constant) in the given ``interval``, False otherwise. 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©rrrrÚÇózis_increasing..©r#©rr rrrrÚ is_increasingŸó(r+cCr$)at Return whether the function is strictly increasing in the given interval. Parameters ========== expression : Expr The target function which is being checked. interval : Set, optional The range of values in which we are testing (defaults to set of all real numbers). symbol : Symbol, optional The symbol present in expression which gets varied over the given range. Returns ======= Boolean True if ``expression`` is strictly increasing in the given ``interval``, False otherwise. 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|dkSr%rr&rrrr'òr(z(is_strictly_increasing..r)r*rrrÚis_strictly_increasingÊr,r-cCr$)a7 Return whether the function is decreasing in the given interval. Parameters ========== expression : Expr The target function which is being checked. interval : Set, optional The range of values in which we are testing (defaults to set of all real numbers). symbol : Symbol, optional The symbol present in expression which gets varied over the given range. Returns ======= Boolean True if ``expression`` is decreasing (either strictly decreasing or constant) in the given ``interval``, False otherwise. Examples ======== >>> from sympy import is_decreasing >>> from sympy.abc import x, y >>> from sympy import S, Interval, oo >>> is_decreasing(1/(x**2 - 3*x), Interval.open(1.5, 3)) True >>> is_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo)) True >>> is_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2)) False >>> is_decreasing(-x**2, Interval(-oo, 0)) False >>> is_decreasing(-x**2 + y, Interval(-oo, 0), x) False cSs|dkSr%rr&rrrr'r(zis_decreasing..r)r*rrrÚ is_decreasingõr,r.cCr$)a Return whether the function is strictly decreasing in the given interval. Parameters ========== expression : Expr The target function which is being checked. interval : Set, optional The range of values in which we are testing (defaults to set of all real numbers). symbol : Symbol, optional The symbol present in expression which gets varied over the given range. Returns ======= Boolean True if ``expression`` is strictly decreasing in the given ``interval``, False otherwise. 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|dkSr%rr&rrrr'Fr(z(is_strictly_decreasing..r)r*rrrÚis_strictly_decreasing s&r/cCsdt|ƒ}|j}|durt|ƒdkrtdƒ‚|p |r| ¡ntdƒ}t| |¡||ƒ}| |¡t j uS)ak Return whether the function is monotonic in the given interval. Parameters ========== expression : Expr The target function which is being checked. interval : Set, optional The range of values in which we are testing (defaults to set of all real numbers). symbol : Symbol, optional The symbol present in expression which gets varied over the given range. Returns ======= Boolean True if ``expression`` is monotonic in the given ``interval``, False otherwise. Raises ====== NotImplementedError Monotonicity check has not been implemented for the queried function. 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 intersectionrr)rr rr!r"Zturning_pointsrrrÚ is_monotonicIs.ÿr1)N)Ú__doc__ZsympyrrZsympy.core.sympifyrZsympy.solvers.solvesetrZsympy.utilities.miscrrrr#r+r-r.r/r1rrrrÚs    X/+++)