Ed-dZddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z mZmZmZddlmZdd Zejd fd Zejd fd Zejd fd Zejd fdZejd fdZejd fdZd 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) filldedentNc6ddlm}||jr tjn tj} tj}|ttttgt tD]6}|jjrt"|jjr|||j||z }7| t(D]}|||jd||z }|S#t"$rt#t-dwxYw)a Find singularities of a given function. Parameters ========== expression : Expr The target function in which singularities need to be found. symbol : Symbol The symbol over the values of which the singularity in expression in being searched for. Returns ======= Set A set of values for ``symbol`` for which ``expression`` has a singularity. An ``EmptySet`` is returned if ``expression`` has no singularities for any given value of ``Symbol``. Raises ====== NotImplementedError Methods for determining the singularities of this function have not been developed. Notes ===== This function does not find non-isolated singularities nor does it find branch points of the expression. Currently supported functions are: - univariate continuous (real or complex) functions References ========== .. [1] https://en.wikipedia.org/wiki/Mathematical_singularity Examples ======== >>> from sympy import singularities, Symbol, log >>> x = Symbol('x', real=True) >>> y = Symbol('y', real=False) >>> singularities(x**2 + x + 1, x) EmptySet >>> singularities(1/(x + 1), x) {-1} >>> singularities(1/(y**2 + 1), y) {-I, I} >>> singularities(1/(y**3 + 1), y) {-1, 1/2 - sqrt(3)*I/2, 1/2 + sqrt(3)*I/2} >>> singularities(log(x), x) {0} rsolvesetNzl Methods for determining the singularities of this function have not been developed.)sympy.solvers.solvesetris_realrReals ComplexesEmptySetrewriterr r r r atomsrexp is_infiniteNotImplementedError is_negativebaserargsr ) expressionsymboldomainrsingsis >DDSII : :Au  *))u  :!&&&999!!#&& 9 9A XXafQi88 8EE ;;;!*.9#:#:;; ;;s CC11'Dcdddlm}t|}|j}|"t |dkrt d|p$|r|ntd}||}||||tj }| |S)a 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. rrNzKThe function has not yet been implemented for all multivariate expressions.x) rrr free_symbolslenrpoprdiffrr is_subset) r predicateintervalrrfreevariable derivativepredicate_intervals r#monotonicity_helperr3ps>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|dkSNrr's r#zis_increasing.. Q!Vr4r3rr.rs r# is_increasingr>sP z+;+;Xv N NNr4c(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|dkSr7r8r9s r#r:z(is_strictly_increasing.. QUr4r<r=s r#is_strictly_increasingrBP z??Hf M MMr4c(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|dkSr7r8r9s r#r:zis_decreasing..#r;r4r<r=s r# is_decreasingrFsX z+;+;Xv N NNr4c(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|dkSr7r8r9s r#r:z(is_strictly_decreasing..NrAr4r<r=s r#is_strictly_decreasingrI&rCr4cRddlm}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 rrNr&zKis_monotonic has not yet been implemented for all multivariate expressions.r') rrrr(r)rr*rr+ intersectionrr)rr.rrr/r0turning_pointss r# is_monotonicrMQs`0/////$$J  "D  #d))a- ! 1   >=$((***&++HXjooh778LLN   0 0AJ >>r4)N)__doc__sympy.core.powerrsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr&sympy.functions.elementary.exponentialr(sympy.functions.elementary.trigonometricrr r r r sympy.utilities.miscr r$rr3r>rBrFrIrMr8r4r#rVs"! 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