a Эм#єz!_expand_delta..csg|]}|ИСqSrrr)┌hrrr%r)┌is_Mulrr┌One┌args┌is_Add┌_has_simple_delta┌func)┌expr┌index┌deltar r)rrr┌ _expand_deltas r$cCsxt||Гsd|fSt|tГr&|tjfS|js4tdГВd}g}|jD]&}|dur^t||Гr^|}qB|а |бqB||j |ОfS)aФ Extract a simple KroneckerDelta from the expression. Explanation =========== Returns the tuple ``(delta, newexpr)`` where: - ``delta`` is a simple KroneckerDelta expression if one was found, or ``None`` if no simple KroneckerDelta expression was found. - ``newexpr`` is a Mul containing the remaining terms; ``expr`` is returned unchanged if no simple KroneckerDelta expression was found. Examples ======== >>> from sympy import KroneckerDelta >>> from sympy.concrete.delta import _extract_delta >>> from sympy.abc import x, y, i, j, k >>> _extract_delta(4*x*y*KroneckerDelta(i, j), i) (KroneckerDelta(i, j), 4*x*y) >>> _extract_delta(4*x*y*KroneckerDelta(i, j), k) (None, 4*x*y*KroneckerDelta(i, j)) See Also ======== sympy.functions.special.tensor_functions.KroneckerDelta deltaproduct deltasummation NzIncorrect expr)r┌ isinstancerrrr┌ ValueErrorr┌_is_simple_delta┌appendr )r!r"r#r┌argrrr┌_extract_delta)s" r*cCsD|аtбr@t||ГrdS|js$|jr@|jD]}t||Гr*dSq*dS)z╪ Returns True if ``expr`` is an expression that contains a KroneckerDelta that is simple in the index ``index``, meaning that this KroneckerDelta is nonzero for a single value of the index ``index``. TF)┌hasrr'rrrr)r!r"r)rrrr\s rcCsBt|tГr>|а|бr>|jd|jdа|б}|r>|абdkSdS)zu Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single value of the index ``index``. rrF)r%rr+rZas_polyZdegree)r#r"┌prrrr'ms r'cCsф|jr|jttt|jГГОS|js&|Sg}g}|jD]4}t|tГr^|а |jd|jdбq4|а |бq4|sr|St |ddН}t|ГdkrРtj St|Гdkrр|dабD]}|а t||d|Гбqи|j|О}||krрt|ГS|S)z0 Evaluate products of KroneckerDelta's. rrTй┌dict)rr ┌list┌map┌_remove_multiple_deltarrr%rr(r┌lenr┌Zero┌keys)r!ZeqsZnewargsr)┌solns┌keyZexpr2rrrr1zs, r1cCsnt|tГrjzLt|jd|jdddН}|rTt|ГdkrTtddД|dабDГОWSWntyhYn0|S)zB Rewrite a KroneckerDelta's indices in its simplest form. rrTr-cSsg|]\}}t||fОСqSr)r)rr6┌valuerrrrбs z#_simplify_delta..)r%rrrr2r┌items┌NotImplementedError)r!Zslnsrrr┌_simplify_deltaШs r:c sИdИddkdkrtjS|аtбs2t|ИГS|jРrjdЙg}t|jtdНD]*}Иdurpt |ИdГrp|ЙqP|а |бqP|j|ОЙtdddН}t ИdtГrюt ИdtГrюtИИГtЗЗЗfd d ДttИdГtИddГГDГГ}nttИИГttИИdИd|dfГИаИd|бtИИd|dИdfГ|ИdИdft ИИdГdН}t|ГSt|ИdГ\Й}ИРs╘t|ИdГ}||kРr╩ztt|ИГГWStРy╚t|ИГYS0t|ИГSt|аИdИdбtИdИdГГtjttИdИddГГS)z┼ Handle products containing a KroneckerDelta. See Also ======== deltasummation sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.products.product щrrTN)r6Zkprime)┌integerc sTg|]L}tИИdИd|dfГИаИd|бtИИd|dИdfГСqS)rrr;)┌deltaproduct┌subs)rZikйr#┌limitZnewexprrrr╞s■ ■z deltaproduct..)┌no_piecewise)rrr+rrr┌sortedrrrr(r r r%┌intr=┌sum┌range┌deltasummationr>r1r*r$r┌AssertionErrorr:)┌fr@rr)┌k┌result┌_┌grr?rr=иsN ¤ ■√ ( r=FcsZИdИddkdkrtjS|аtбs2t|ИГSИd}t||Г}|jrjt|jЗЗfddД|j DГОГSt ||Г\}}|dur└|jdur└|j\}}Иd|dkdkr└Иd|dkdkr└dЙ|s╬t|ИГSt|j d|j d|Г} t | Гdkr·tjSt | ГdkРrt|ИГS| d} ИРr,|а|| бSt|а|| бtИddЕОа| бftjdfГS) au Handle summations containing a KroneckerDelta. Explanation =========== The idea for summation is the following: - If we are dealing with a KroneckerDelta expression, i.e. KroneckerDelta(g(x), j), we try to simplify it. If we could simplify it, then we sum the resulting expression. We already know we can sum a simplified expression, because only simple KroneckerDelta expressions are involved. If we couldn't simplify it, there are two cases: 1) The expression is a simple expression: we return the summation, taking care if we are dealing with a Derivative or with a proper KroneckerDelta. 2) The expression is not simple (i.e. KroneckerDelta(cos(x))): we can do nothing at all. - If the expr is a multiplication expr having a KroneckerDelta term: First we expand it. If the expansion did work, then we try to sum the expansion. If not, we try to extract a simple KroneckerDelta term, then we have two cases: 1) We have a simple KroneckerDelta term, so we return the summation. 2) We didn't have a simple term, but we do have an expression with simplified KroneckerDelta terms, so we sum this expression. Examples ======== >>> from sympy import oo, symbols >>> from sympy.abc import k >>> i, j = symbols('i, j', integer=True, finite=True) >>> from sympy.concrete.delta import deltasummation >>> from sympy import KroneckerDelta >>> deltasummation(KroneckerDelta(i, k), (k, -oo, oo)) 1 >>> deltasummation(KroneckerDelta(i, k), (k, 0, oo)) Piecewise((1, i >= 0), (0, True)) >>> deltasummation(KroneckerDelta(i, k), (k, 1, 3)) Piecewise((1, (i >= 1) & (i <= 3)), (0, True)) >>> deltasummation(k*KroneckerDelta(i, j)*KroneckerDelta(j, k), (k, -oo, oo)) j*KroneckerDelta(i, j) >>> deltasummation(j*KroneckerDelta(i, j), (j, -oo, oo)) i >>> deltasummation(i*KroneckerDelta(i, j), (i, -oo, oo)) j See Also ======== deltaproduct sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.sums.summation r;rrTcsg|]}t|ИИГСqSr)rF)rrйr@rArrr3rz"deltasummation..Nщ)rr3r+rrr$rrr rr*Zdelta_rangerr2rr>r rZ as_relational)rHr@rA┌xrLr#r!ZdinfZdsupr5r7rrMrrFфs:D ( ■rFN)F)!┌__doc__ZproductsrZ summationsrrZ sympy.corerrrr Zsympy.core.cacher Zsympy.core.sortingrZsympy.functionsrr rZsympy.polys.polytoolsrZsympy.sets.setsrZsympy.solvers.solversrr$r*rr'r1r:r=rFrrrr┌s2 2 ;