+ i |^RIHtHtHt^RIHtHt^RIHt^RI H t ^RI H t ^RI Ht^RIHt!RR ]4tR #) )Soodiff)DefinedFunctionArgumentIndexError) fuzzy_not)Eq)im) Piecewise) Heavisidecfa]tRt^toRtRtR Rlt]R4tRt Rt Rt R R lt ] t ] tR tVtR#) 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 Tc NV^8XdVPwr#pV\P\P\R4\R439dVP W#V^, 4#VP 'd W@P W#V^, 4,#R#\ W4h)a 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. N)argsrZero NegativeOnefunc is_positiver)selfargindexxans&& k/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/functions/special/singularity_functions.pyfdiffSingularityFunction.fdiffXs q=iiGA!QVVQ]]AbE1R599yyqs++11---%T4 4c RTpTpTpWE, p\\V4P4'd \R4h\\V4P4'd \R4hV\P JgV\P Jd\P #V^,P 'd \R4hVP'd\P#VP'dDVP'd\PV,#VP'd Wv,#V\PRRR39dOVP 'gVP'd\P#VP'd\#R#R#)a@ 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.Nrr)rr is_zero ValueErrorrNaN is_negativeis_extended_negativeris_nonnegativeis_extended_nonnegativeris_extended_positiver)clsvariableoffsetexponentrrrshifts&&&& revalSingularityFunction.evalqs-V    RY&& ' 'WX X RU]] # #]^ ^ AEE>Q!%%Z55L E   `a a  % % %66M   }}}vvqy ,,,x B+ +   E$>$>$>vv }}}  ,rc <VPwr4pV\P\R4\R4\R439d#\\\ W4, ^43R4#VP 'd%\W4, V,W4, ^83R4#R#)zF Converts a Singularity Function expression into its Piecewise form. Nrrr!)rT)rrrr rr r'rrkwargsrrrs&*, r_eval_rewrite_as_Piecewise.SingularityFunction._eval_rewrite_as_Piecewises{ ))a "quae4 4b"QUA,/; ;    quqj!%1*5yA Arc lVPwr4pVR8Xd5\\W4, 4VPP 4^4#VR8Xd5\\W4, 4VPP 4^4#VR8Xd5\\W4, 4VPP 4^4#VR8Xd5\\W4, 4VPP 4^4#VP 'd'W4, V,\W4, ^4,#R#)zO Rewrites a Singularity Function expression using Heavisides and DiracDeltas. Nr!rr)rrr free_symbolspopr'r2s&*, r_eval_rewrite_as_Heaviside.SingularityFunction._eval_rewrite_as_Heavisides ))a 7 !%(!..*<*<*>B B 7 !%(!..*<*<*>B B 7 !%(!..*<*<*>B B 7 !%(!..*<*<*>B B   EA:iq11 1 rchVPwrEpWE, PV^4pV^8d\P#VP'd:VP'd(VR8Xd\P#\P #VP 'd Wv,#\P#)rr7)rsubsrrr"Oner)rrlogxcdirzrrr.s&&&& r_eval_as_leading_term)SingularityFunction._eval_as_leading_termsw))a Q" q566M YYY5===!RZ166 2QUU 2    8Ovv rNcVPwrVpWV, PV^4pV^8d\P#VP'd:VP'd(VR8Xd\P#\P #VP 'd!WV, V,PWW4R7#\P#)r)r?r@r7)rr=rrr"r>r _eval_nseries)rrrr?r@rArr.s&&&&& rrE!SingularityFunction._eval_nseriess))a Q" q566M YYY5===!RZ166 2QUU 2    UQJ--a-I Ivv r))Nr)__name__ __module__ __qualname____firstlineno____doc__is_realr classmethodr/r4r:rBrE_eval_rewrite_as_DiracDelta$_eval_rewrite_as_HeavisideDiracDelta__static_attributes____classdictcell__) __classdict__s@rrrsRENG52BBH B2$  #=+E(rrN) sympy.corerrrsympy.core.functionrrsympy.core.logicrsympy.core.relationalr $sympy.functions.elementary.complexesr $sympy.functions.elementary.piecewiser 'sympy.functions.special.delta_functionsr rrGrrr\s-""C&$3:=]F/]Fr