+ i&Rt^RIHt^RIHtHt^RIHtHtH t H t ^RI H t ^RI Ht^RIHtHtHt^RIHt^RIHt^R IHt] R 4t] R 4t] R 4t] R 4t] R4t] R4t] R4t] RRl4t R#)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)solvecVP'gV#Rp\p\P.pVPFypVf\VP 'dJ\ WQ4'd9RpVPpVPUu.uFqd^,V,NK ppKbVUu.uF qfV,NK ppK{ V!V!#uupiuupi)z: Expand the first Add containing a simple KroneckerDelta. NT)is_MulrrOneargsis_Add_has_simple_deltafunc)exprindexdeltartermshts&& R/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/concrete/delta.py _expand_deltars ;;; E D UUGE YY =QXXX*;A*E*EE66D)*0A1XaZZE0E"'(%QqSS%E(E  <1(s B= CcZ\W4'gRV3#\V\4'dV\P3#VP 'g \ R4hRp.pVPF-pVf\WA4'dTpKVPV4K/ W P!V!3#)a8 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)rrrrargs&& r_extract_deltar&)sD T ) )d|$''aee} ;;;)** E Eyy =-c99E LL   99e$ %%cJaVP\4'd\VS4'dR#VP'gVP'dO\ ;QJd)V3RlVP 4F 'gK R# R#!V3RlVP 44#R#)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``. Tc3<<"TFp\VS4xK R#5i)N)r).0r%rs& r $_has_simple_delta..gsJ (e44 sF)hasr r#rranyr)rrs&frrr\si xx D% ( ( ;;;$+++3J J33 J3 J3J JJ J r'c \V\4'dmVPV4'dVVP^,VP^,, P V4pV'dVP 4^8H#R#)zi Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single value of the index ``index``. F)r!r r-ras_polydegree)rrps&& rr#r#ksZ %((UYYu-=-= ZZ]UZZ] * 3 3E : 88:? " r'c VP'd2VP!\\\VP 44!#VP 'gV#.p.pVP Ffp\V\4'd=VPVP ^,VP ^,, 4KUVPV4Kh V'gV#\VRR7p\V4^8Xd\P#\V4^8Xd\Y$^,P4UUu.uFwrV\WV4NK upp, pVP!V!pW8wd \ V4#V#uuppi)z( Evaluate products of KroneckerDelta's. Tdict)rrlistmap_remove_multiple_deltarrr!r r$rlenrZeroitems)reqsnewargsr%solnskvexpr2s& rr8r8xs  {{{yy$s#9499EFGG ;;; CGyy c> * * JJsxx{SXXa[0 1 NN3    #D !E 5zQvv Uq1X^^5EF5ETQN1(5EFF 7# =)%0 0 K Gs3E6c x\V\4'd\VP^,VP^,, RR7pV'dK\ V4^8Xd;\ V^,P 4UUu.uFwr#\W#3!NK upp!#V#V#uuppi \dT#i;i)z: Rewrite a KroneckerDelta's indices in its simplest form. Tr4)r!r rrr9rr;NotImplementedError)rslnskeyvalues& r_simplify_deltarGs $'' 1 ! 44@DD Q.21gmmo?.= ,c\:.=?@@ K4K ?"   K s$A-B*B$ B*$B** B98B9c Zaaa S^,S^,, ^8R8Xd\P#VP\4'g \ VS4#VP 'EdRo.p\ VP\R7F5pSf\VS^,4'dVoK$VPV4K7 VP!V!o \RRR7p\S^,\4'dx\S^,\4'd[\S S4\!VVV 3Rl\#\S^,4\S^,^,4444,pM\S S4\%\S S^,S^,V^, 34SP'S^,V4,\S S^,V^,S^,34,VS^,S^,3\S S^,4R7,p\)V4#\+VS^,4wopS'g<\-VS^,4pW8wd\/\VS44#\ VS4#\)VP'S^,S^,4\S^,S^,4,4\P\3\S^,S^,^, 44,,# \0d\TS4u#i;i)z Handle products containing a KroneckerDelta. See Also ======== deltasummation sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.products.product TN)rEkprime)integerc 3<"TFpp\SS^,S^,V^, 34SPS^,V4,\SS^,V^,S^,34,xKr R#5i)N) deltaproductsubs)r*ikrlimitnewexprs& rr+deltaproduct..sz8Nu9EWuUVxY^_`YacehiciNj8k 58R(9)WuQxaq&BC9D9DNusA8A;) no_piecewise)rrr-r rrsortedrr rr$rrr!intrMsumrangedeltasummationrNr8r&rrAssertionErrorrG) frPrr%r?result_grrQs &f @@rrMrMs qE!H !d*uu 55 q%  xxx!&&&67C}!23a!A!A S! 8 &&%. (D ) eAh $ $E!Hc)B)B!'51C8NSTWX]^_X`TacfglmngorsgsctNu85F "'51NWuQxq1q5&AB 58Q'(WuQxQa&ABCE!HeAh'.waA 5F&f--aq*HE1  !U1X & 6 .l1e455q%  !!&&q58"<^ERSHV[\]V^=_"_ ` onU1XuQx!|DEE FF " .#Au-- .s#LL*)L*c V^,V^,, ^8R8Xd\P#VP\4'g \ W4#V^,p\ W4pVP 'd>\VP!VPUu.uFp\WQV4NK up!4#\WC4wrgVeOVPeAVPwrV^,V, ^8*R8XdV^,V , ^8R8XdRpV'g \ W4#\VP^,VP^,, V4p \V 4^8Xd\P#\V 4^8wd \W4#V ^,p V'dVP!W;4#\#VP!W;4\%VR,!P'V 43\PR34#uupi)a 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 T:N)rr:r-r rrrr rrrXr& delta_rangerr9rrNr r as_relational) rZrPrSxr]rrrdinfdsupr>rFs &&& rrXrXsH qE!H !d*vv 55 "" aAaAxxx FFQVVLV^Al;VL MO O!&KE  1 1 =&&  !HtOq T )uQx$!/C.LL "" %**Q-%**Q-/ 3E 5zQvv Uq1} !HEyy""  1 hc 3AA%HI  +MsH N)F)!__doc__productsr summationsrr sympy.corerrrrsympy.core.cacher sympy.core.sortingr sympy.functionsr r r sympy.polys.polytoolsrsympy.sets.setsrsympy.solvers.solversrrr&rr#r8rGrMrXr'rrqs&))$/EE($'  & /& /&d          8     7F 7Ft f fr'