o 8Vaw'@sdZddlmZmZmZmZddlmZddlm Z ddl m Z m Z m Z ddlmZeddZed d Zed d Zed dZeddZeddZeddZedddZdS)z This module implements sums and products containing the Kronecker Delta function. References ========== .. [1] http://mathworld.wolfram.com/KroneckerDelta.html )AddMulSDummy)cacheit)default_sort_key)KroneckerDelta Piecewisepiecewise_fold)Intervalcsx|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)termsr 6/usr/lib/python3/dist-packages/sympy/concrete/delta.py z!_expand_delta..csg|]}|qSr r r )hr rr!s)is_MulrrOneargsis_Add_has_simple_deltafunc)exprindexdeltarr )rrr _expand_deltas rcCsxt||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 isinstancerrrr ValueErrorr_is_simple_deltaappendr)rrrrargr r r_extract_delta%s "    r$cCsD|tr t||r dS|js|jr |jD] }t||rdSqdS)z Returns True if ``expr`` is an expression that contains a KroneckerDelta that is simple in the index ``index``, meaning that this KroneckerDelta is nonzero for a single value of the index ``index``. TF)hasrr!rrrr)rrr#r r rrXs     rcCsBt|tr||r|jd|jd|}|r|dkSdS)zu Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single value of the index ``index``. rF)rrr%rZas_polyZdegree)rrpr r rr!is  r!cCsddlm}|jr|jttt|jS|js|Sg}g}|jD]}t |t r5| |jd|jdq | |q |s?|S||dd}t |dkrNt jSt |dkrv|dD]}| t ||d|qZ|j|}||krvt|S|S)z0 Evaluate products of KroneckerDelta's. rsolver&Tdict) sympy.solversr)rrlistmap_remove_multiple_deltarrrrr"lenrZerokeys)rr)ZeqsZnewargsr#solnskeyZexpr2r r rr/vs.        r/cCsddlm}t|tr@z*||jd|jddd}|r0t|dkr3tdd|dDWSW|SW|Sty?Y|Sw|S)zB Rewrite a KroneckerDelta's indices in its simplest form. rr(r&Tr*cSsg|] \}}t||fqSr )r)rr4valuer r rrsz#_simplify_delta..) r,r)rrrr0ritemsNotImplementedError)rr)Zslnsr r r_simplify_deltas"     r8c s4ddlm}dddkdkrtjS|ts||S|jrdg}t|jt dD]}dur=t |dr=|q-| |q-|j |t ddd }tdtrtdtrttfd d ttdtddD}t|Stttdd|dfd|td|ddf|ddft dd }t|St|d\}st|d}||krdd lm} z| t|WStyt|YSw||St|ddtddtjttdddS)z Handle products containing a KroneckerDelta. See Also ======== deltasummation sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.products.product r)productr&TN)r4Zkprime)Zintegerc sTg|]&}tdd|dfd|td|ddfqS)rr&r:) deltaproductsubs)rZikrlimitZnewexprr rrsz deltaproduct..) no_piecewise)factor)Zsympy.concrete.productsr9rrr%rrsortedrrrr"rrrintr;sumrangedeltasummationr<r/r$rZsympyr@AssertionErrorr8) fr>r9rr#kresult_gr@r r=rr;sV            (r;Fcszddlm}ddlm}dddkdkrtjS|ts%||Sd}t||}|j rAt |j fdd|j DSt ||\}}|d url|jd url|j\} } d| dkdkrld| dkdkrld|ss||S||j d|j d|} t| dkrtjSt| dkrdd lm} | |S| d} r||| St||| tdd | ftjdfS) 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) summationr(r:r&Tcsg|]}t|qSr )rE)rrr>r?r rr7rz"deltasummation..N)Sum)Zsympy.concrete.summationsrLr,r)rr1r%rrrr rrr$Z delta_ranger0rNr<r r Z as_relational)rGr>r?rLr)xrKrrZdinfZdsupr3rNr5r rMrrEs@ D     (       rEN)F)__doc__Z sympy.corerrrrZsympy.core.cacherZsympy.core.compatibilityrZsympy.functionsrr r Z sympy.setsr rr$rr!r/r8r;rEr r r rs,     2     >