o 8Va[ @sxddlmZmZmZmZddlmZmZddlm Z ddl m Z ddl m Z ddlmZddlmZGdd d eZd S) )Ssympifyoodiff)FunctionArgumentIndexError) fuzzy_not)Eq)im) Piecewise) Heavisidec@sVeZdZdZdZdddZeddZdd Zd d Z dddZ dddZ e Z e Z d S)SingularityFunctionaL 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)`` 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), (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) + DiracDelta(x + 3) - DiracDelta(x, 1) >>> expr.rewrite(DiracDelta) (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1) >>> expr.rewrite('HeavisideDiracDelta') (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1) See Also ======== DiracDelta, Heaviside References ========== .. [1] https://en.wikipedia.org/wiki/Singularity_function TcCs~|dkr:t|jd}t|jd}t|jd}|dks!|dkr*||||dS|jr8|||||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. rrN)rargsfunc is_positiver)selfZargindexxanrO/usr/lib/python3/dist-packages/sympy/functions/special/singularity_functions.pyfdiffXs zSingularityFunction.fdiffcCst|}t|}t|}||}tt|jrtdtt|jr&td|tjus0|tjur3tjS|djr>> 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(x, a, n).eval(3, 5, 1) 0 >>> SingularityFunction(x, a, n).eval(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.rzASingularity Functions are not defined for exponents less than -2.rN)rrr is_zero ValueErrorrZNaNZ is_negativeZis_extended_negativeZerois_nonnegativeZis_extended_nonnegativeZis_extended_positiveZInfinity)clsvariableoffsetZexponentrrrshiftrrrevalss.-    zSingularityFunction.evalcOsr|jd}|jd}t|jd}|dks|dkr%ttt||dfdS|jr7t|||||dkfdSdS)zV Converts a Singularity Function expression into its Piecewise form. rrrrr)rTN)rrr rr rrrkwargsrrrrrr_eval_rewrite_as_Piecewises  z.SingularityFunction._eval_rewrite_as_PiecewisecOs|jd}|jd}t|jd}|dkr"tt|||jdS|dkr3tt|||jdS|jrB|||t||SdS)z_ Rewrites a Singularity Function expression using Heavisides and DiracDeltas. rrrrrN)rrrr Z free_symbolspoprr%rrr_eval_rewrite_as_Heavisides  z.SingularityFunction._eval_rewrite_as_HeavisideNrcCs^|j\}}}|||d}|dkrtjS|jr%|jr%|dkr"tjStjS|jr,||StjS)Nrr)rsubsrrrOner)rrlogxcdirzrrr#rrr_eval_as_leading_terms  z)SingularityFunction._eval_as_leading_termcCsp|j\}}}|||d}|dkrtjS|jr%|jr%|dkr"tjStjS|jr5|||j||||dStjS)Nrr)r,r-)rr*rrrr+r _eval_nseries)rrrr,r-r.rr#rrrr0s  z!SingularityFunction._eval_nseries)r)Nr)__name__ __module__ __qualname____doc__Zis_realr classmethodr$r'r)r/r0Z_eval_rewrite_as_DiracDeltaZ$_eval_rewrite_as_HeavisideDiracDeltarrrrr sG  C   r N)Z sympy.corerrrrZsympy.core.functionrrZsympy.core.logicrZsympy.core.relationalr Z$sympy.functions.elementary.complexesr Z$sympy.functions.elementary.piecewiser Z'sympy.functions.special.delta_functionsr r rrrrs