Ed&@dZddlmZddlmZmZddlmZmZm Z m Z ddl m Z ddl mZddlmZmZmZdd lmZdd lmZdd lmZe d Ze d Ze dZe dZe dZe dZe dZe ddZ dS)z This module implements sums and products containing the Kronecker Delta function. References ========== .. [1] http://mathworld.wolfram.com/KroneckerDelta.html )product)Sum summation)AddMulSDummy)cacheit)default_sort_key)KroneckerDelta Piecewisepiecewise_fold)factor)Interval)solvec|js|Sd}t}tjg|jD]F|4jr-t |rd}j}fdjD8fdDG|S)zB Expand the first Add containing a simple KroneckerDelta. NTc&g|] }d|zS)r).0ttermss 4lib/python3.11/site-packages/sympy/concrete/delta.py z!_expand_delta..#s!000AU1XaZ000cg|]}|zSrr)rrhs rrz!_expand_delta..%s(((QQqS(((r)is_Mulrr Oneargsis_Add_has_simple_deltafunc)exprindexdeltar#rrs @@r _expand_deltar's ; E D UGE Y))  )QX )*;Au*E*E )E6D0000000EE((((%(((EE 4<rc$t||sd|fSt|tr|tjfS|jst dd}g}|jD],}|t||r|}| |-||j |fS)a Extract a simple KroneckerDelta from the expression. Explanation =========== Returns the tuple ``(delta, newexpr)`` where: - ``delta`` is a simple KroneckerDelta expression if one was found, or ``None`` if no simple KroneckerDelta expression was found. - ``newexpr`` is a Mul containing the remaining terms; ``expr`` is returned unchanged if no simple KroneckerDelta expression was found. Examples ======== >>> from sympy import KroneckerDelta >>> from sympy.concrete.delta import _extract_delta >>> from sympy.abc import x, y, i, j, k >>> _extract_delta(4*x*y*KroneckerDelta(i, j), i) (KroneckerDelta(i, j), 4*x*y) >>> _extract_delta(4*x*y*KroneckerDelta(i, j), k) (None, 4*x*y*KroneckerDelta(i, j)) See Also ======== sympy.functions.special.tensor_functions.KroneckerDelta deltaproduct deltasummation NzIncorrect expr) r" isinstancer r rr ValueErrorr _is_simple_deltaappendr#)r$r%r&rargs r_extract_deltar.)sD T5 ) )d|$''ae} ;+)*** E Ey  -c599 EE LL     949e$ %%rc|tr=t||rdS|js|jr|jD]}t ||rdSdS)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+r!rr r")r$r%r-s rr"r"\sv xx D% ( ( 4 ; $+ y  $S%00 44 5rct|tr]||rH|jd|jdz |}|r|dkSdS)zu Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single value of the index ``index``. rrF)r)r r0r as_polydegree)r&r%ps rr+r+msl %((#UYYu-=-=# Z]UZ] * 3 3E : :  #88::? " 5rc|jr/|jttt|jS|js|Sg}g}|jD][}t|tr/| |jd|jdz F| |\|s|St|d}t|dkr tj St|dkrk|dD]1}| t||d|2|j|}||krt |S|S)z0 Evaluate products of KroneckerDelta's. rrTdict)r!r#listmap_remove_multiple_deltar rr)r r,rlenr Zerokeys)r$eqsnewargsr-solnskeyexpr2s rr:r:zsc  {Hty$s#949EEFFGG ; CGy   c> * * JJsx{SXa[0 1 1 1 1 NN3      #D ! ! !E 5zzQ1v Uq18==?? ? ?C NN>#uQx}== > > > > 7# 5= 1)%00 0 Krc(t|tr| t|jd|jdz d}|r>t |dkr+t d|dDSn#t$rYnwxYw|S)zB Rewrite a KroneckerDelta's indices in its simplest form. rrTr6c*g|]\}}t||fSr)r )rrAvalues rrz#_simplify_delta..s5??? *U,c5\:???r)r)r rr r;ritemsNotImplementedError)r$slnss r_simplify_deltarIs $'' 1 ! 44@@@D @D Q @??.21gmmoo???@@"    D  KsA)B BBc H ddz dkdkr tjS|tst |S|jrdg}t |jtD]2}t|dr|| |3|j | tdd}tdtrtdtrft t! fd t#tdtddzDz}nt t%t dd|dz fd|zt d|dzdfz|ddft d z}t)|St+|d\}sjt-|d}||kr> t/t|S#t0$rt|cYSwxYwt |St)|ddtddztjt3tdddz zzS) z Handle products containing a KroneckerDelta. See Also ======== deltasummation sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.products.product rrTN)rAkprime)integerc g|]d}tdd|dz fd|ztd|dzdfzeS)rrrK) deltaproductsubs)rikr&limitnewexprs rrz deltaproduct..s9w9w9wIKWuQxq26&BCC 58R(()WuQxaq&BCCD9w9w9wr) no_piecewise)r rr0r rr!sortedr r r"r,r#r r)intrOsumrangedeltasummationrPr:r.r'rAssertionErrorrI) frRrr-kresult_gr&rSs ` @@rrOrOs_ qE!H !d*u 55 !q%   x.!&&6777 " "C "!23a!A!A " S!!!!!&%. (D ) ) ) eAh $ $ E!Hc)B)B !'511C9w9w9w9w9w9wOTTWX]^_X`TaTacfglmngorsgsctctNuNu9w9w9w55FF "'511NWuQxq1q5&ABB 58Q''(WuQxQa&ABBCE!HeAh'.waAA 555F&f---aq**HE1 ! !U1X & & 6 . .l1e44555! . . .#Au----- .q%   !!&&q58"<"<^ERSHV[\]V^=_=_"_ ` ` onU1XuQx!|DDEEE FFsI..J  J Fcddz dkdkr tjS|tst |Sd}t ||}|jr)t|jfd|j DSt||\}}|9|j 2|j \}}d|z dkdkrd|z dkdkrd|st |St|j d|j dz |} t| dkr tjSt| dkrt|S| d} r||| St!||| t#dd| 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 rKrrTc2g|]}t|Sr)rY)rrrRrTs rrz"deltasummation..3s%LLL^Aul;;LLLrN)r r<r0r rr'r!rr#r r. delta_rangerr;rrPrr as_relational) r[rRrTxr_r&r$dinfdsupr@rEs `` rrYrYsH qE!H !d*v 55 #E""" aAaAxO AFLLLLLQVLLL MOO O!A&&KE4   1 & d !HtOq T ) uQx$!/C.L L #E""" %*Q-%*Q-/ 3 3E 5zzQv Uq1e}} !HE#yyE"""  1e  hac 3AA%HHI   rN)F)!__doc__productsr summationsrr sympy.corerrr r sympy.core.cacher sympy.core.sortingr sympy.functionsr rrsympy.polys.polytoolsrsympy.sets.setsrsympy.solvers.solversrr'r.r"r+r:rIrOrYrrrrrs&&&&&&&&))))))))))))$$$$$$//////EEEEEEEEEE(((((($$$$$$''''''  & /&/& /&d             :      8F8F 8Fv fff fffr