g ddlmZmZmZddlmZmZddlmZddl m Z ddl m Z ddl mZddlmZGdd eZd S) )Soodiff)FunctionArgumentIndexError) fuzzy_not)Eq)im) Piecewise) HeavisidecXeZdZdZdZd dZedZdZdZ d d Z d d Z e Z e Z dS)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 Tc0|dkr|j\}}}|tjtjtdtdfvr||||dz S|jr|||||dz zSdSt ||)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. rN)argsrZero NegativeOnefunc is_positiver)selfargindexxans i/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/functions/special/singularity_functions.pyfdiffzSingularityFunction.fdiffXs q==iGAq!QVQ]AbEE1R55999yyAqs+++ .1a1---- . .%T844 4cN|}|}|}||z }tt|jrtdtt|jrtd|tjus|tjur tjS|dzjrtd|jr tjS|j r"|jrtj|zS|j r||zS|tj dddfvr(|js|j r tjS|jr tSdSdS) aP Returns a simplified form or a value of Singularity Function depending on the argument passed by the object. Explanation =========== The ``eval()`` method is automatically called when the ``SingularityFunction`` class is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words, ``eval()`` method is not needed to be called explicitly, it is being called and evaluated once the object is called. Examples ======== >>> 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)clsvariableoffsetexponentrrrshifts revalzSingularityFunction.evalqsCV   Q RYY& ' ' YWXX X RUU] # # _]^^ ^ AE>>Q!%ZZ5L E  b`aa a  % 6M  } !vqy , ax B+ + +  E$> v }  , +  rc*|j\}}}|tjtdtdtdfvr(ttt ||z dfdS|jrt||z |z||z dkfdSdS)zV Converts a Singularity Function expression into its Piecewise form. rrr"r)rTN)rrrr rr r(rrkwargsrrrs r_eval_rewrite_as_Piecewisez.SingularityFunction._eval_rewrite_as_Piecewises )1a "quuaee4 4 4b"QUA,,/;; ;   Bq1uqj!a%1*5yAA A B BrcR|j\}}}|dkr8tt||z |jdS|dkr8tt||z |jdS|dkr8tt||z |jdS|dkr8tt||z |jdS|jr||z |zt||z dzSd S) z_ Rewrites a Singularity Function expression using Heavisides and DiracDeltas. r"r!rrrN)rrr free_symbolspopr(r2s r_eval_rewrite_as_Heavisidez.SingularityFunction._eval_rewrite_as_Heavisides )1a 77 !a%((!.*<*<*>*>BB B 77 !a%((!.*<*<*>*>BB B 77 !a%((!.*<*<*>*>BB B 77 !a%((!.*<*<*>*>BB B  2EA:iAq111 1 2 2rNrc|j\}}}||z |d}|dkr tjS|jr%|jr|dkr tjn tjS|jr||zStjS)Nrr8)rsubsrrr#Oner)rrlogxcdirzrrr/s r_eval_as_leading_termz)SingularityFunction._eval_as_leading_termsz)1aQ Q"" q556M Y 5= !RZZ166QU 2   !8Ov rc*|j\}}}||z |d}|dkr tjS|jr%|jr|dkr tjn tjS|jr||z |z||||StjS)Nrr8)r?r@)rr=rrr#r>r _eval_nseries)rrrr?r@rArr/s rrDz!SingularityFunction._eval_nseriess)1aQ Q"" q556M Y J5= J!RZZ166QU 2   JUQJ--aD-II Iv r)r)Nr)__name__ __module__ __qualname____doc__is_realr classmethodr0r4r;rBrD_eval_rewrite_as_DiracDelta$_eval_rewrite_as_HeavisideDiracDeltarrrrsEENG55552BB[BH B B B222$        #=+E(((rrN) sympy.corerrrsympy.core.functionrrsympy.core.logicrsympy.core.relationalr $sympy.functions.elementary.complexesr $sympy.functions.elementary.piecewiser 'sympy.functions.special.delta_functionsr rrMrrrUs""""""""""<<<<<<<<&&&&&&$$$$$$333333::::::======]F]F]F]F]F(]F]F]F]F]Fr