o o h&@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]'|dur.jr.t|r.d}j}fddjDqfddDq|S)zB Expand the first Add containing a simple KroneckerDelta. NTcsg|]}d|qS)r.0t)termsrh/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/concrete/delta.py #z!_expand_delta..csg|]}|qSrrr)hrrr%s)is_MulrrOneargsis_Add_has_simple_deltafunc)exprindexdeltar r)rrr _expand_deltas r$cCsxt||s d|fSt|tr|tjfS|jstdd}g}|jD]}|dur/t||r/|}q!| |q!||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||r dS|js|jr |jD] }t||rdSqdS)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+ras_polydegree)r#r"prrrr'ms  r'cCs|jr|jttt|jS|js|Sg}g}|jD]}t|tr/| |jd|jdq| |q|s9|St |dd}t |dkrHt j St |dkrp|dD]}| t||d|qT|j|}||krpt|S|S)z0 Evaluate products of KroneckerDelta's. rrTdict)rr listmap_remove_multiple_deltarrr%r r(rlenrZerokeys)r!eqsnewargsr)solnskeyexpr2rrrr3zs,       r3cCsxt|tr:z*t|jd|jddd}|r*t|dkr-tdd|dDWSW|SW|Sty9Y|Sw|S)zB Rewrite a KroneckerDelta's indices in its simplest form. rrTr/cSsg|] \}}t||fqSr)r )rr:valuerrrrsz#_simplify_delta..)r%r rrr4ritemsNotImplementedError)r!slnsrrr_simplify_deltas    r@c sdddkdkrtjS|tst|S|jrdg}t|jtdD]}dur7t |dr7|q'| |q'|j |t ddd}t dtryt dtryttfd d ttdtddD}t|Stttdd|dfd|td|ddf|ddft dd }t|St|d\}st|d}||krztt|WStyt|YSwt|St|ddtddtjttdddS) z Handle products containing a KroneckerDelta. See Also ======== deltasummation sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.products.product rrTN)r:kprime)integerc3sX|]'}tdd|dfd|td|ddfVqdS)rrrAN) deltaproductsubs)rikr#limitnewexprrr s zdeltaproduct..) no_piecewise)rrr+r rrsortedrr rr(r r r%intrDsumrangedeltasummationrEr3r*r$rAssertionErrorr@)frHrr)kresult_grrGrrDsR           (rDFc sVdddkdkrtjS|tst|Sd}t||}|jr5t|jfdd|j DSt ||\}}|dur`|j dur`|j \}}d|dkdkr`d|dkdkr`d|sgt|St |j d|j d|} t | dkr}tjSt | 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 rArrTcsg|]}t|qSr)rP)rrrHrKrrr2rz"deltasummation..N)rr5r+r rr$rrr rr* delta_rangerr4rrEr r as_relational) rRrHrKxrVr#r!dinfdsupr9r<rrWrrPs:D    (      rPN)F)!__doc__productsr summationsrr sympy.corerrrr sympy.core.cacher sympy.core.sortingr sympy.functionsr r rsympy.polys.polytoolsrsympy.sets.setsrsympy.solvers.solversrr$r*rr'r3r@rDrPrrrrs4        2     :