ó ¡¼™\c@ s¸ddlmZmZddlmZmZmZmZddlm Z m Z ddl m Z ddl mZddlmZddlmZddlmZd e fd „ƒYZd S( iÿÿÿÿ(tprint_functiontdivision(tStsympifytootdiff(tFunctiontArgumentIndexError(t fuzzy_not(tEq(tim(t Piecewise(t HeavisidetSingularityFunctioncB sMeZdZeZdd„Zed„ƒZd„Zd„Z e Z e Z RS(s The Singularity functions are a class of discontinuous functions. They 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 Reference ========= .. [1] https://en.wikipedia.org/wiki/Singularity_function icC s®|dkr›t|jdƒ}t|jdƒ}t|jdƒ}|dks]|dkrt|j|||dƒS|jrª||j|||dƒSnt||ƒ‚dS(s( Returns the first derivative of a DiracDelta Function. The difference between ``diff()`` and ``fdiff()`` is:- ``diff()`` is the user-level function and ``fdiff()`` is an object method. ``fdiff()`` is just 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. iiiiÿÿÿÿN(Rtargstfunct is_positiveR(tselftargindextxtatn((sLlib/python2.7/site-packages/sympy/functions/special/singularity_functions.pytfdiffUs  cC s-t|ƒ}t|ƒ}t|ƒ}||}tt|ƒjƒrRtdƒ‚ntt|ƒjƒrvtdƒ‚n|tjks”|tjkr›tjS|djr·tdƒ‚n|jrÇtjS|j rå|j rå|||S|dksý|dkr)|js|j rtjS|jr)tj SndS(sm Returns a simplified form or a value of Singularity Function depending on the argument passed by the object. 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(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 s8Singularity Functions are defined only for Real Numbers.s>Singularity Functions are not defined for imaginary exponents.isASingularity Functions are not defined for exponents less than -2.iÿÿÿÿiþÿÿÿN( RRR tis_zerot ValueErrorRtNaNt is_negativetZerotis_nonnegativeRtInfinity(tclstvariabletoffsettexponentRRRtshift((sLlib/python2.7/site-packages/sympy/functions/special/singularity_functions.pytevalls*(        cO s£|jd}|jd}t|jdƒ}|dksE|dkrkttt||dƒfdtfƒS|jrŸt|||||dkfdtfƒSdS(sV Converts a Singularity Function expression into its Piecewise form. iiiiÿÿÿÿiþÿÿÿN(RRR RR tTrueR(RRtkwargsRRR((sLlib/python2.7/site-packages/sympy/functions/special/singularity_functions.pyt_eval_rewrite_as_Piecewise«s  & cO s²|jd}|jd}t|jdƒ}|dkr\tt||ƒ|jjƒdƒS|dkr‹tt||ƒ|jjƒdƒS|jr®|||t||ƒSdS(s_ Rewrites a Singularity Function expression using Heavisides and DiracDeltas. iiiiþÿÿÿiÿÿÿÿN(RRRR t free_symbolstpopR(RRR%RRR((sLlib/python2.7/site-packages/sympy/functions/special/singularity_functions.pyt_eval_rewrite_as_Heaviside¹s   # # ( t__name__t __module__t__doc__R$tis_realRt classmethodR#R&R)t_eval_rewrite_as_DiracDeltat$_eval_rewrite_as_HeavisideDiracDelta(((sLlib/python2.7/site-packages/sympy/functions/special/singularity_functions.pyR sA ?  N(t __future__RRt sympy.coreRRRRtsympy.core.functionRRtsympy.core.logicRtsympy.core.relationalR t$sympy.functions.elementary.complexesR t$sympy.functions.elementary.piecewiseR t'sympy.functions.special.delta_functionsR R (((sLlib/python2.7/site-packages/sympy/functions/special/singularity_functions.pyts"