+ j/pRt^RIHt^RIHt^RIHt^RIHt^RI H t ^RI H t H t HtHtHt^RIHtHtHtHtHtHtHtHtHt^RIHtRR lt]P<R 3R lt]P<R 3R lt ]P<R 3R lt!]P<R 3Rlt"]P<R 3Rlt#]P<R 3Rlt$R #)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) filldedentNc^RIHpVf3VP'd\PM\P p\P pVP\\\\.\4pVP\\\\ .\"4pVP%\&4F\pVP(P*'d\,hVP(P.'gKCWC!VP0W4, pK^ VP%\2\4\64F#pWC!VP8^,W4, pK% VP%\:\<4FQpWC!VP8^,^, W4, pWC!VP8^,^,W4, pKS V# \,d\-\?R44hi;i)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} solvesetzl Methods for determining the singularities of this function have not been developed.) sympy.solvers.solvesetris_realrReals ComplexesEmptySetrewriterrr r r r r rrratomsrexp is_infiniteNotImplementedError is_negativebaserrrargsrrr) expressionsymboldomainrsingseis&&& ]/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/calculus/singularities.py singularitiesr-sgx0 ~"NNN ;    S#s3S 9 IItT4. 5Auu   ))uu   !&&&99  !!#ue4A XaffQi8 8E5!!%/A XaffQi!mV< 0$J  " "D ~ t9q=%5  >>$((*&+H*J!)J"7177K   0 11c\VRW4#)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 V^8#r/s&r,is_increasing..Q!Vr<r;r&r6r's&&&r, is_increasingrHsP z+;X NNr<c\VRW4#)a 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 V^8#r?rArBs&r,rC(is_strictly_increasing..QUr<rFrGs&&&r,is_strictly_increasingrMP z?H MMr<c\VRW4#)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 V^8*#r?rArBs&r,rCis_decreasing..+rEr<rFrGs&&&r, is_decreasingrRsX z+;X NNr<c\VRW4#)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(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 V^8#r?rArBs&r,rC(is_strictly_decreasing..VrLr<rFrGs&&&r,is_strictly_decreasingrV.rNr<cF^RIHp\V4pVPpVf\ V4^8d \ R4hT;'g$V'dVP 4M \R4pV!VPV4WQ4pVPV4\PJ#)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 rzKis_monotonic has not yet been implemented for all multivariate expressions.r/) rrrr0r1r"r2rr3 intersectionrr)r&r6r'rr7r8turning_pointss&&& r, is_monotonicrZYs`0$J  " "D ~#d)a-! 1  >>$((*&+Hjooh7LN   0AJJ >>r<)N)%__doc__sympy.core.powerrsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr&sympy.functions.elementary.exponentialr(sympy.functions.elementary.trigonometricrrr r r %sympy.functions.elementary.hyperbolicr r rrrrrrrsympy.utilities.miscrr-rr;rHrMrRrVrZrAr<r,rds"!"$&6LL>>>+S;v9:.2b()wwt(OV12(NV()wwt,O^12(NV'(ggd=?r<