g/&dZddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z mZmZmZddlmZmZmZmZmZmZmZmZmZdd lmZdd Zejd fd Zejd fd Z ejd fdZ!ejd fdZ"ejd fdZ#ejd fdZ$d S)af Singularities ============= This module implements algorithms for finding singularities for a function and identifying types of functions. The differential calculus methods in this module include methods to identify the following function types in the given ``Interval``: - Increasing - Strictly Increasing - Decreasing - Strictly Decreasing - Monotonic )Pow)S)Symbol)sympify)log)seccsccottancos) sechcschcothtanhcoshasechacschatanhacoth) filldedentNcvddlm}||jr tjn tj} tj}|ttttgt}|tttt gt"}|t&D]6}|jjrt,|jjr|||j||z }7|t2t4t6D]}|||jd||z }|t:t<D]>}|||jddz ||z }|||jddz||z }?|S#t,$rt-t?dwxYw)a Find singularities of a given function. Parameters ========== expression : Expr The target function in which singularities need to be found. symbol : Symbol The symbol over the values of which the singularity in expression in being searched for. Returns ======= Set A set of values for ``symbol`` for which ``expression`` has a singularity. An ``EmptySet`` is returned if ``expression`` has no singularities for any given value of ``Symbol``. Raises ====== NotImplementedError Methods for determining the singularities of this function have not been developed. Notes ===== This function does not find non-isolated singularities nor does it find branch points of the expression. Currently supported functions are: - univariate continuous (real or complex) functions References ========== .. [1] https://en.wikipedia.org/wiki/Mathematical_singularity Examples ======== >>> from sympy import singularities, 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} rsolvesetNzl Methods for determining the singularities of this function have not been developed.) sympy.solvers.solvesetris_realrReals ComplexesEmptySetrewriterr r r r r rrrratomsrexp is_infiniteNotImplementedError is_negativebaserrrargsrrr) expressionsymboldomainrsingseis X/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/calculus/singularities.py singularitiesr/sx0///// ~"N; ;    S#s3S 9 9 IItT4. 5 5 : :Au  *))u  :!&&&999!!#ue44 9 9A XXafQi88 8EE!!%// = =A XXafQi!mVV<< 0/////$$J  "D ~ t99q==%5  >=$((***&++H**J!))J"7"717KK   0 1 11c(t|d||S)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 c|dkSNrr1s r.zis_increasing.. Q!Vr>r=r(r8r)s r. is_increasingrHsP z+;+;Xv N NNr>c(t|d||S)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 c|dkSrArBrCs r.rDz(is_strictly_increasing.. QUr>rFrGs r.is_strictly_increasingrLP z??Hf M MMr>c(t|d||S)a 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(S(3)/2, 3)) True >>> 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(1/(x**2 - 3*x), Interval.Ropen(-oo, 1.5)) False >>> is_decreasing(-x**2, Interval(-oo, 0)) False >>> is_decreasing(-x**2 + y, Interval(-oo, 0), x) False c|dkSrArBrCs r.rDzis_decreasing..+rEr>rFrGs r. is_decreasingrPsX z+;+;Xv N NNr>c(t|d||S)aZ 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(1/(x**2 - 3*x), Interval.Ropen(-oo, 1.5)) False >>> is_strictly_decreasing(-x**2, Interval(-oo, 0)) False >>> is_strictly_decreasing(-x**2 + y, Interval(-oo, 0), x) False c|dkSrArBrCs r.rDz(is_strictly_decreasing..VrKr>rFrGs r.is_strictly_decreasingrS.rMr>cRddlm}t|}|j}|"t |dkrt d|p$|r|ntd}|||||}| |tj uS)a 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(S(3)/2, 3)) True >>> 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 rrNrzKis_monotonic has not yet been implemented for all multivariate expressions.r1) rrrr2r3r$r4rr5 intersectionrr)r(r8r)rr9r:turning_pointss r. is_monotonicrWYs`0/////$$J  "D ~#d))a--! 1   >=$((***&++HXjooh778LLN   0 0AJ >>r>)N)%__doc__sympy.core.powerrsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr&sympy.functions.elementary.exponentialr(sympy.functions.elementary.trigonometricrr r r r %sympy.functions.elementary.hyperbolicr rrrrrrrrsympy.utilities.miscrr/rr=rHrLrPrSrWrBr>r.ras"! 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