U Zʹj&@sdZddlmZddlmZmZddlmZmZm Z m Z ddl m Z ddl mZddlmZmZmZdd lmZdd lmZdd lmZe d d Ze ddZe ddZe ddZe ddZe ddZe ddZe dddZ dS)z This module implements sums and products containing the Kronecker Delta function. References ========== .. [1] https://mathworld.wolfram.com/KroneckerDelta.html )product)Sum summation)AddMulSDummy)cacheit)default_sort_key)KroneckerDelta Piecewisepiecewise_fold)factor)Interval)solvecsx|js |Sd}t}tjg|jD]N|dkr\jr\t|r\d}j}fddjDq fddDq |S)zB Expand the first Add containing a simple KroneckerDelta. NTcsg|]}d|qS)r.0t)termsrS/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/concrete/delta.py #sz!_expand_delta..csg|] }|qSrrr)hrrr%s)is_MulrrOneargsis_Add_has_simple_deltafunc)exprindexdeltarr)rrr _expand_deltas r#cCsxt||sd|fSt|tr&|tjfS|js4tdd}g}|jD]&}|dkr^t||r^|}qB| |qB||j |fS)a 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 isinstancer rrr ValueErrorr_is_simple_deltaappendr)r r!r"rargrrr_extract_delta)s"     r)cCsD|tr@t||rdS|js$|jr@|jD]}t||r*dSq*dS)z Returns True if ``expr`` is an expression that contains a KroneckerDelta that is simple in the index ``index``, meaning that this KroneckerDelta is nonzero for a single value of the index ``index``. TF)hasr r&rrrr)r r!r(rrrr\s     rcCsBt|tr>||r>|jd|jd|}|r>|dkSdS)zu Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single value of the index ``index``. rrF)r$r r*rZas_polyZdegree)r"r!prrrr&ms  r&cCs|jr|jttt|jS|js&|Sg}g}|jD]4}t|tr^| |jd|jdq4| |q4|sr|St |dd}t |dkrt j St |dkr|dD]}| t||d|q|j|}||krt|S|S)z0 Evaluate products of KroneckerDelta's. rrTdict)rrlistmap_remove_multiple_deltarrr$r r'rlenrZerokeys)r ZeqsZnewargsr(solnskeyZexpr2rrrr0zs,       r0cCspt|trlzLt|jd|jddd}|rTt|dkrTtdd|dDWSWntk rjYnX|S)zB Rewrite a KroneckerDelta's indices in its simplest form. rrTr,cSsg|]\}}t||fqSr)r )rr5valuerrrrsz#_simplify_delta..)r$r rrr1ritemsNotImplementedError)r Zslnsrrr_simplify_deltas  r9c s dddkdkrtjS|ts2t|S|jrjdg}t|jtdD]*}dkrpt |drp|qP| |qP|j |t ddd}t dtrt dtrttfd d ttdtddD}nttttdd|dfd|td|ddf|ddft dd }t|St|d\}st|d}||krztt|WStk rt|YSXt|St|ddtddtjttdddS) z Handle products containing a KroneckerDelta. See Also ======== deltasummation sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.products.product rrTN)r5Zkprime)integerc3sV|]N}tdd|dfd|td|ddfVqdS)rrr:N) deltaproductsubs)rikr"limitZnewexprrr s zdeltaproduct..) no_piecewise)rrr*r rrsortedrr rr'rr r$intr<sumrangedeltasummationr=r0r)r#rAssertionErrorr9)fr@rr(kresult_grr?rr<sN           (r<Fc sZdddkdkrtjS|ts2t|Sd}t||}|jrjt|jfdd|j DSt ||\}}|dk r|j dk r|j \}}d|dkdkrd|dkdkrd|st|St |j d|j d|} t | dkrtjSt | dkrt|S| d} r,||| St||| tdd| ftjdfS) aw 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 could not 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 did not 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|qSr)rG)rrr@rBrrr2sz"deltasummation..N)rr2r*r rr#rrrrr)Z delta_rangerr1rr=r rZ as_relational) rIr@rBxrMr"r ZdinfZdsupr4r6rrNrrGs:D    (     rGN)F)!__doc__ZproductsrZ summationsrrZ sympy.corerrrr Zsympy.core.cacher Zsympy.core.sortingr Zsympy.functionsr r rZsympy.polys.polytoolsrZsympy.sets.setsrZsympy.solvers.solversrr#r)rr&r0r9r<rGrrrrs2        2     :