B [&@sdZddlmZmZddlmZmZmZmZddl m Z ddl m Z m Z ddlmZmZmZddlmZe dd Ze d d Ze d d Ze ddZe ddZe ddZe ddZe dddZdS)z This module implements sums and products containing the Kronecker Delta function. References ========== - http://mathworld.wolfram.com/KroneckerDelta.html )print_functiondivision)AddMulSDummy)cacheit)default_sort_keyrange)KroneckerDelta Piecewisepiecewise_fold)Intervalcs~|js |Sd}t}tdgxX|jD]N|dkr`jr`t|r`d}j}fddjDq$fddDq$W|S)zB Expand the first Add containing a simple KroneckerDelta. NTcsg|]}d|qS)r).0t)termsr3lib/python3.7/site-packages/sympy/concrete/delta.py sz!_expand_delta..csg|] }|qSrr)rr)hrrr"s)is_Mulrrargsis_Add_has_simple_deltafunc)exprindexdeltarr)rrr _expand_deltas  rcCs~t||sd|fSt|tr(|tdfS|js6tdd}g}x0|jD]&}|dkrbt||rb|}qF||qFW||j |fS)as Extract a simple KroneckerDelta from the expression. 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 NrzIncorrect expr) r isinstancer rr ValueErrorr_is_simple_deltaappendr)rrrrargrrr_extract_delta&s    r%cCsF|trBt||rdS|js$|jrBx|jD]}t||r,dSq,WdS)z Returns True if ``expr`` is an expression that contains a KroneckerDelta that is simple in the index ``index``, meaning that this KroneckerDelta is nonzero for a single value of the index ``index``. TF)hasr r"rrrr)rrr$rrrrVs     rcCsBt|tr>||r>|jd|jd|}|r>|dkSdS)zu Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single value of the index ``index``. rrF)r r r&rZas_polyZdegree)rrprrrr"gs  r"cCsddlm}|jr(|jttt|jS|js2|Sg}g}x>|jD]4}t |t rl| |jd|jdqB| |qBW|s|S||dd}t |dkrt jSt |dkrx,|dD]}| t ||d|qW|j|}||krt|S|S)z0 Evaluate products of KroneckerDelta's. r)solverT)dict) sympy.solversr(rrlistmap_remove_multiple_deltarrr r r#lenrZerokeys)rr(ZeqsZnewargsr$solnskeyZexpr2rrrr-ts.       r-cCszddlm}t|trvyJ||jd|jddd}|r^t|dkr^tdd|dDSWntk rtYnX|S)zB Rewrite a KroneckerDelta's indices in its simplest form. r)r(rT)r)cSsg|]\}}t||fqSr)r )rr2valuerrrrsz#_simplify_delta..) r*r(r r rr.ritemsNotImplementedError)rr(Zslnsrrr_simplify_deltas  r6c sXddlm}dddkdkr*tjS|ts>||S|jrzdg}x.) no_piecewise)factor)Eq)Zsympy.concrete.productsr7rZOner&r rsortedrr rr#rrr intr9sumr deltasummationr:r-r%rZsympyr>AssertionErrorr?r6) fr;r7rr$kresult_gr>r?cr)rr;r<rr9sL      ( *     (r9Fc s&ddlm}ddlm}dddkdkr6tjS|tsJ||Sd}t||}|j rt |j fdd|j DSt ||\}}|s||S||j d|j d|} t| dkrtjSt| dkrt|S| d} r||| St||| tdd | ftjdfS) a_ Handle summations containing a KroneckerDelta. 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, Piecewise >>> 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) summation)r(r8rTcsg|]}t|qSr)rC)rr)r;r=rrr4sz"deltasummation..)Zsympy.concrete.summationsrKr*r(rr/r&r rrr rrr%r.ZSumr:r rZ as_relational) rEr;r=rKr(xrIrrr1r3r)r;r=rrCs2A           rCN)F)__doc__Z __future__rrZ sympy.corerrrrZsympy.core.cacherZsympy.core.compatibilityr r Zsympy.functionsr r r Z sympy.setsrrr%rr"r-r6r9rCrrrr s    0    A