\hJU4SSKJr SSKJr SSKJr SSKJrJr SSK J r SSK J r J r SSKJr SSKJrJr SS KJr SS KJrJr SS KJr SS KJr SS KJrJr SSKJ r SSK!J"r" SSK#J$r$ SSK%J&r&J'r' SSK(J)r) Sr*SS.Sjr+"SS\5r,"SS\,5r-g))Add)Tuple)Expr) AppliedUndefUndefinedFunction)Mul)Equality Relational)S)SymbolDummy)sympify)piecewise_fold Piecewise)BooleanFunction) MatrixBase)IntervalSet)Range)Idx)flatten)sift is_sequence)sympy_deprecation_warningc[U5n[U[5(ao[USU06upVU(a[ SU55(d [ SSSSS9 UR nURn[U"U/UQ70UD6U"U/UQ70UD65$U[RLa[R$U(aX[USU06upV[U5H;up[U 5S:XdMURU S U S 5n[U S S 6XY'M= OFURn [U 5S :wa[S U-5eU V s/sHn [U 5PM sn S peU[!U5:Xa5[#UR$5U-nUR&nU[!U5:XaM50n UV s1sHoS iM nn UR)[*5H%nUR,"U6(aM[/5X'M' UR1U 5n[3U5nUR1U R55VVs0sH unnUU_M snn5nXU4$s sn fs sn fs snnf)zReturn either a special return value or the tuple, (function, limits, orientation). This code is common to both ExprWithLimits and AddWithLimits.discretec3># UHn[U5S:Hv M g7fNlen).0limits W/var/www/auris/envauris/lib/python3.13/site-packages/sympy/concrete/expr_with_limits.py _common_new.. sCF5s5zQFsax Creating a indefinite integral with an Eq() argument is deprecated. This is because indefinite integrals do not preserve equality due to the arbitrary constants. If you want an equality of indefinite integrals, use Eq(Integral(a, x), Integral(b, x)) explicitly. z1.6z!deprecated-indefinite-integral-eq)deprecated_since_versionactive_deprecations_target stacklevelrNzspecify dummy variables for %s)r isinstancer _process_limitsallrlhsrhsr NaN enumerater!subsr free_symbols ValueErrortypelistlimitsfunctionatomsrhasr xreplaceritems)clsr;rsymbols assumptionsr: orientationr1r2ilifreesrepssymbols_of_integrationpkvs r$ _common_newrMs>x H(H%%.wJJ3CFCCC %*/+N llllC9'9[9C9'9[9; ;155uu -wJJv&EA2w!|#==A27!2cr7O ' $$ t9>08;= =156AuQx6  h hoo&/$$ h  D,23FqdF3 ^^I &uu,--gDG'  &Hh'H  4::>*Q"4"41v{z!A$44GadH--'00aDHHadhh.AabE!e,,#'001B1BQqTYYBQw!#!"Q%7%77!"1!bg]BtGbL)I IA!"1!QZZ"QqTB)G GA.-5)::7:: #A!A$ ..'!A$ U2S2SaD q6Q;t|!(8#r) " !AB%&I%Qq%&IIA1v!)S11TQYqy eQi0 qy%i55&/__ioo%#%>$qtrz:J:J*45\*]$]%#%>$qtrz:J:J*45]*^$^ eQi0 qyTQY1Q4< eI&67  eIt&<= 3c'lBCCgj  K'J&$-% $% $-% $%s01R;R$R%$R) R&%R&) R65R6c\rSrSrSrSr\S5r\S5r\S5r \S5r \S5r \S 5r \S 5r S rS r\S 5r\S5rSrg)ExprWithLimits)is_commutativec\SSKJn [X/UQ7S[X50UD6n[ U[ 5(aUupnOU$[ SU55(a [S5e[R"U40UD6nU/n U RU5 [ U 5Ul URUl U$)Nr)Productrc3V# UHn[U5S:g=(d SU;v M! g7frr r"ls r$r%)ExprWithLimits.__new__..s$8As1v{'dai's')z:ExprWithLimits requires values for lower and upper bounds.) sympy.concrete.productsrnrM issubclassr.tupleanyr7r__new__extend_argsrl) r@r;rArBrnprer:_objarglists r$rwExprWithLimits.__new__s3#>'>->1<> c5 ! !"% HaJ 88 8 8YZ Zll3.+.*v'N %44 c URS$)zReturn the function applied across limits. Examples ======== >>> from sympy import Integral >>> from sympy.abc import x >>> Integral(x**2, (x,)).function x**2 See Also ======== limits, variables, free_symbols rryselfs r$r;ExprWithLimits.functions"zz!}rc.URR$N)r;kindrs r$rExprWithLimits.kinds}}!!!rc URSS$)zReturn the limits of expression. Examples ======== >>> from sympy import Integral >>> from sympy.abc import x, i >>> Integral(x**i, (i, 1, 3)).limits ((i, 1, 3),) See Also ======== function, variables, free_symbols r-Nrrs r$r:ExprWithLimits.limitss"zz!"~rcJURVs/sHoSPM sn$s snf)a1Return a list of the limit variables. >>> from sympy import Sum >>> from sympy.abc import x, i >>> Sum(x**i, (i, 1, 3)).variables [i] See Also ======== function, limits, free_symbols as_dummy : Rename dummy variables sympy.integrals.integrals.Integral.transform : Perform mapping on the dummy variable r)r:rrqs r$ variablesExprWithLimits.variables s" #kk*k!k***s cnURVs/sHn[U5S:wdMUSPM sn$s snf)apReturn only variables that are dummy variables. Examples ======== >>> from sympy import Integral >>> from sympy.abc import x, i, j, k >>> Integral(x**i, (i, 1, 3), (j, 2), k).bound_symbols [i, j] See Also ======== function, limits, free_symbols as_dummy : Rename dummy variables sympy.integrals.integrals.Integral.transform : Perform mapping on the dummy variable r-r)r:r!rs r$ bound_symbolsExprWithLimits.bound_symbolss0&#kk9kSVq[!k999s2 2czURURp!URVs0sH-nUSUSRUS1:XaUSO [5_M/ nnUR U5nURnUHinXFSn[ U5S:XaUR U5 M,Xu;aURU5 USSHnURUR5 M Mk UR5VVs0sHupXx_M nnnUV s1sHoRX5iM sn $s snfs snnfs sn f)z This method returns the symbols in the object, excluding those that take on a specific value (i.e. the dummy symbols). Examples ======== >>> from sympy import Sum >>> from sympy.abc import x, y >>> Sum(x, (x, y, 1)).free_symbols {y} rr-N) r;r:r6r r>r!addremoveupdater?get) rr;r:rDrHisymsxabrLrKr{s r$r6ExprWithLimits.free_symbols1s"" ==$++&[[" !ad//AaD69aduwF  "$$T*%%CV A3x1} ! z QW Q^^,"&..(-.1..!"/.s4D-;D2D8c$UR(+$)z7Return True if the Sum has no free symbols, else False.)r6rs r$ is_numberExprWithLimits.is_numberWs$$$$rcURVs/sHoDSU:waUOXU4PM nnURnUR"U/UQ76$s snf)Nr)r:r;func)rxabrDr: integrands r$_eval_intervalExprWithLimits._eval_interval\sN;?;;G;a! 1ay0;GMM yy,V,,HsAc UR[UR5pCUR5 [ U[ 5(a+UR RUR 5(GaSn[U5HupgS[U5:Xa XS:XaUR(aU4nOX4n[US/USSVs/sHoRX5PM snQ76XF'[USR RUR 55S:wdMSn O [ U[[45(a[UR 5R[UR#[ 555n [UR 5R[UR$55n U R'U 5(d [)S5eSnU(aUR+X5nOb[U5HSupg[U5S:XdM[US/USSVs/sHoRX5PM snQ76XF'XS:XdMS O [U5HBupg[U5S:XdMUSUS- R,(dM2[US5XF'MD UR5 UR."U/UQ76$s snfs snf) ao Perform substitutions over non-dummy variables of an expression with limits. Also, can be used to specify point-evaluation of an abstract antiderivative. Examples ======== >>> from sympy import Sum, oo >>> from sympy.abc import s, n >>> Sum(1/n**s, (n, 1, oo)).subs(s, 2) Sum(n**(-2), (n, 1, oo)) >>> from sympy import Integral >>> from sympy.abc import x, a >>> Integral(a*x**2, x).subs(x, 4) Integral(a*x**2, (x, 4)) See Also ======== variables : Lists the integration variables transform : Perform mapping on the dummy variable for integrals change_index : Perform mapping on the sum and product dummy variables Tr-rNFz-substitution cannot create dummy dependenciesrrP)r;r9r:reverser.r r6 intersectionr4r!rOr_subsrrsetrr<argsissubsetr7r5is_zeror) roldnewrr: sub_into_funcrDrrqsy2sy1s r$ _eval_subsExprWithLimits._eval_subsasM6}}d4;;&7f #v&&  --d.?.?@@ M#F+C=SF]}}"f"j!#a&Ps12w+Ow!GGC,=w+OP s1v**778H8HIJaO$)M,# .?@AA$..)66s399V;L7MN$..)66s388}E||C(($GII $ yy*$F+s8q= %c!f T3qr7/S7a0A7/S TFI!f} ,  'FA3x1}#a&3q6/!:!:!:!#a&O ( yy'''9,P"0Ts K "K c SnURHtn[U5S:XaN[SUSS55(a g[SUSS55(aUSRcSnM\M^M`USRbMrSnMv U(agg) a{ Returns True if the limits are known to be finite, either by the explicit bounds, assumptions on the bounds, or assumptions on the variables. False if known to be infinite, based on the bounds. None if not enough information is available to determine. Examples ======== >>> from sympy import Sum, Integral, Product, oo, Symbol >>> x = Symbol('x') >>> Sum(x, (x, 1, 8)).has_finite_limits True >>> Integral(x, (x, 1, oo)).has_finite_limits False >>> M = Symbol('M') >>> Sum(x, (x, 1, M)).has_finite_limits >>> N = Symbol('N', integer=True) >>> Product(x, (x, 1, N)).has_finite_limits True See Also ======== has_reversed_limits Frc38# UHoRv M g7fr is_infiniterps r$r%3ExprWithLimits.has_finite_limits..s6g}}gr-Nc3<# UHoRSLv M g7frrrps r$r%rs@1$.srT)r:r!rvr)rret_Nonelims r$has_finite_limits ExprWithLimits.has_finite_limitssB;;C3x1}6c!"g666 @AB@@@1v))1#'2A q6%%-#H rcSnURHEn[U5S:Xa3Uup4nXT- nUR(a gUR(aMASnME g U(agg)aH Returns True if the limits are known to be in reversed order, either by the explicit bounds, assumptions on the bounds, or assumptions on the variables. False if known to be in normal order, based on the bounds. None if not enough information is available to determine. Examples ======== >>> from sympy import Sum, Integral, Product, oo, Symbol >>> x = Symbol('x') >>> Sum(x, (x, 8, 1)).has_reversed_limits True >>> Sum(x, (x, 1, oo)).has_reversed_limits False >>> M = Symbol('M') >>> Integral(x, (x, 1, M)).has_reversed_limits >>> N = Symbol('N', integer=True, positive=True) >>> Sum(x, (x, 1, N)).has_reversed_limits False >>> Product(x, (x, 2, N)).has_reversed_limits >>> Product(x, (x, 2, N)).subs(N, N + 2).has_reversed_limits False See Also ======== sympy.concrete.expr_with_intlimits.ExprWithIntLimits.has_empty_sequence FrTN)r:r!is_extended_negativeis_extended_nonnegative)rrrvarrrdifs r$has_reversed_limits"ExprWithLimits.has_reversed_limitss^J;;C3x1} e++00#H rN)__name__ __module__ __qualname____firstlineno__ __slots__rwpropertyr;rr:rrr6rrrrr__static_attributes__rrr$rjrjs#I,$""$++"::(#/#/J%%- M(^00d33rrjc@\rSrSrSrSrSrSrSrSr Sr S r Sr g ) AddWithLimitsizNRepresents unevaluated oriented additions. Parent class for Integral and Sum. rcSSKJn [X/UQ7S[X50UD6n[ U[ 5(aUupnOU$[ R"U40UD6nXq-/n U RU5 [ U 5Ul URUl U$)Nr)Sumr) sympy.concrete.summationsrrMrtr.rurrwrxryrl) r@r;rArBrrzr:rCr|r}s r$rwAddWithLimits.__new__!s1#:':):-8: c5 ! !,/ )HkJll3.+.'(v'N %44 rc[S[UR555(a5UR"URR 5/URQ76$g)Nc38# UHoRv M g7fris_realr"rs r$r%.AddWithLimits._eval_adjoint..37"6Qyy"6r)r0rr:rr;adjointrs r$ _eval_adjointAddWithLimits._eval_adjoint2sD 7'$++"67 7 799T]]224Ct{{C Crc[S[UR555(a5UR"URR 5/URQ76$g)Nc38# UHoRv M g7frrrs r$r%0AddWithLimits._eval_conjugate..8rr)r0rr:rr; conjugaters r$_eval_conjugateAddWithLimits._eval_conjugate7D 7'$++"67 7 799T]]446EE Erc[S[UR555(a5UR"URR 5/URQ76$g)Nc38# UHoRv M g7frrrs r$r%0AddWithLimits._eval_transpose..=rr)r0rr:rr; transposers r$_eval_transposeAddWithLimits._eval_transpose<rrc ^S[TR5:Xa|TRR"S0UD6nUR(aL[ UR U4Sj5n[US6TR"[US6/TRQ76-$T$TR"TR/TRSSQ76R5nURTRS5(d1TRSTRS/5R5U-$[U[5(a-TRUTRS5R5$T$)Nr-cx>UR=(a' [TR5UR-(+$r)rlrrr6)wrs r$,AddWithLimits._eval_factor..Es/13C3C4A/!..@@4ArTFrr,r) r!r:r;factoris_Mulrrrrr=rdoitr.)rhintssummandouts` r$ _eval_factorAddWithLimits._eval_factorAs/ DKK mm**3U3G~~7<<*ABCItyyc%j1A("[[(""" ii B Ab0ABIIKG;;t~~b122yyT[[_$56;;=gEEGS))yy$++b/:AACC rc ^TRR"S0UD6nURSS5nUR(aeU(d UR(aMTR SLa>[ URVs/sH nTR"U/TRQ76PM" sn6$[U[5(aURU4Sj5$UTR:waTR"U/TRQ76$T$s snf)NforceFc>>TR"U/TRQ76$r)rr:)rrs r$r2AddWithLimits._eval_expand_basic..Xstyy/IT[[/Irr) r;expandris_Addrlrrrrr:r.r applyfunc)rrrrrDs` r$_eval_expand_basic AddWithLimits._eval_expand_basicQs--&&// '5) NN)?)?''u4W\\J\13t{{3\JK K  , ,$$%IJ J  %99W3t{{3 3 Ks;'C?N) rrrr__doc__rrwrrrrrrrrr$rrs,I"    rr).sympy.core.addrsympy.core.containersrsympy.core.exprrsympy.core.functionrrsympy.core.mulrsympy.core.relationalr r sympy.core.singletonr sympy.core.symbolr r sympy.core.sympifyr$sympy.functions.elementary.piecewiserrsympy.logic.boolalgrsympy.matrices.matrixbasersympy.sets.setsrrsympy.sets.fancysetsrsympy.tensor.indexedrsympy.utilitiesrsympy.utilities.iterablesrrsympy.utilities.exceptionsrrMr/rjrrrr$rsm' ?6"+&/0)&$#7@D)N(,gTPTPf ANAr