o o h @stddlmZmZmZddlmZmZddlmZddl m Z ddl m Z ddl mZddlmZGdd d eZd S) )Soodiff)FunctionArgumentIndexError) fuzzy_not)Eq)im) Piecewise) Heavisidec@sVeZdZdZdZdddZeddZdd Zd d Z dddZ dddZ e Z e Z d S)SingularityFunctionaU Singularity functions are a class of discontinuous functions. Explanation =========== Singularity functions take a variable, an offset, and an exponent as arguments. These functions are represented using Macaulay brackets as: SingularityFunction(x, a, n) := ^n The singularity function will automatically evaluate to ``Derivative(DiracDelta(x - a), x, -n - 1)`` if ``n < 0`` and ``(x - a)**n*Heaviside(x - a, 1)`` if ``n >= 0``. Examples ======== >>> from sympy import SingularityFunction, diff, Piecewise, DiracDelta, Heaviside, Symbol >>> from sympy.abc import x, a, n >>> SingularityFunction(x, a, n) SingularityFunction(x, a, n) >>> y = Symbol('y', positive=True) >>> n = Symbol('n', nonnegative=True) >>> SingularityFunction(y, -10, n) (y + 10)**n >>> y = Symbol('y', negative=True) >>> SingularityFunction(y, 10, n) 0 >>> SingularityFunction(x, 4, -1).subs(x, 4) oo >>> SingularityFunction(x, 10, -2).subs(x, 10) oo >>> SingularityFunction(4, 1, 5) 243 >>> diff(SingularityFunction(x, 1, 5) + SingularityFunction(x, 1, 4), x) 4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4) >>> diff(SingularityFunction(x, 4, 0), x, 2) SingularityFunction(x, 4, -2) >>> SingularityFunction(x, 4, 5).rewrite(Piecewise) Piecewise(((x - 4)**5, x >= 4), (0, True)) >>> expr = SingularityFunction(x, a, n) >>> y = Symbol('y', positive=True) >>> n = Symbol('n', nonnegative=True) >>> expr.subs({x: y, a: -10, n: n}) (y + 10)**n The methods ``rewrite(DiracDelta)``, ``rewrite(Heaviside)``, and ``rewrite('HeavisideDiracDelta')`` returns the same output. One can use any of these methods according to their choice. >>> expr = SingularityFunction(x, 4, 5) + SingularityFunction(x, -3, -1) - SingularityFunction(x, 0, -2) >>> expr.rewrite(Heaviside) (x - 4)**5*Heaviside(x - 4, 1) + DiracDelta(x + 3) - DiracDelta(x, 1) >>> expr.rewrite(DiracDelta) (x - 4)**5*Heaviside(x - 4, 1) + DiracDelta(x + 3) - DiracDelta(x, 1) >>> expr.rewrite('HeavisideDiracDelta') (x - 4)**5*Heaviside(x - 4, 1) + DiracDelta(x + 3) - DiracDelta(x, 1) See Also ======== DiracDelta, Heaviside References ========== .. [1] https://en.wikipedia.org/wiki/Singularity_function TcCsl|dkr1|j\}}}|tjtjtdtdfvr!||||dS|jr/|||||dSdSt||)aK Returns the first derivative of a DiracDelta Function. Explanation =========== The difference between ``diff()`` and ``fdiff()`` is: ``diff()`` is the user-level function and ``fdiff()`` is an object method. ``fdiff()`` is a convenience method available in the ``Function`` class. It returns the derivative of the function without considering the chain rule. ``diff(function, x)`` calls ``Function._eval_derivative`` which in turn calls ``fdiff()`` internally to compute the derivative of the function. r N)argsrZero NegativeOnefunc is_positiver)selfargindexxanr/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/functions/special/singularity_functions.pyfdiffXs  zSingularityFunction.fdiffcCs|}|}|}||}tt|jrtdtt|jr td|tjus*|tjur-tjS|djr6td|jr>> from sympy import SingularityFunction, Symbol, nan >>> from sympy.abc import x, a, n >>> SingularityFunction(x, a, n) SingularityFunction(x, a, n) >>> SingularityFunction(5, 3, 2) 4 >>> SingularityFunction(x, a, nan) nan >>> SingularityFunction(x, 3, 0).subs(x, 3) 1 >>> SingularityFunction(4, 1, 5) 243 >>> x = Symbol('x', positive = True) >>> a = Symbol('a', negative = True) >>> n = Symbol('n', nonnegative = True) >>> SingularityFunction(x, a, n) (-a + x)**n >>> x = Symbol('x', negative = True) >>> a = Symbol('a', positive = True) >>> SingularityFunction(x, a, n) 0 z8Singularity Functions are defined only for Real Numbers.z>Singularity Functions are not defined for imaginary exponents.zASingularity Functions are not defined for exponents less than -4.rrN)rr is_zero ValueErrorrNaN is_negativeis_extended_negativeris_nonnegativeis_extended_nonnegativeris_extended_positiver)clsvariableoffsetexponentrrrshiftrrrevalqs4+   zSingularityFunction.evalcOsj|j\}}}|tjtdtdtdfvr!ttt||dfdS|jr3t|||||dkfdSdS)zV Converts a Singularity Function expression into its Piecewise form. rrrr)rTN)rrrr rrr$rrkwargsrrrrrr_eval_rewrite_as_Piecewises z.SingularityFunction._eval_rewrite_as_PiecewisecOs|j\}}}|dkrtt|||jdS|dkr(tt|||jdS|dkr9tt|||jdS|dkrJtt|||jdS|jrZ|||t||dSd S) z_ Rewrites a Singularity Function expression using Heavisides and DiracDeltas. rrrrr N)rrr free_symbolspopr$r-rrr_eval_rewrite_as_Heavisides z.SingularityFunction._eval_rewrite_as_HeavisideNrcCs^|j\}}}|||d}|dkrtjS|jr%|jr%|dkr"tjStjS|jr,||StjS)Nrr2)rsubsrrrOner)rrlogxcdirzrrr+rrr_eval_as_leading_terms  z)SingularityFunction._eval_as_leading_termcCsp|j\}}}|||d}|dkrtjS|jr%|jr%|dkr"tjStjS|jr5|||j||||dStjS)Nrr2)r8r9)rr6rrrr7r _eval_nseries)rrrr8r9r:rr+rrrr<s  z!SingularityFunction._eval_nseries)r )Nr)__name__ __module__ __qualname____doc__is_realr classmethodr,r/r5r;r<_eval_rewrite_as_DiracDelta$_eval_rewrite_as_HeavisideDiracDeltarrrrr sG  D   r N) sympy.corerrrsympy.core.functionrrsympy.core.logicrsympy.core.relationalr$sympy.functions.elementary.complexesr $sympy.functions.elementary.piecewiser 'sympy.functions.special.delta_functionsr r rrrrs