ó ¡¼™\c@ sjddlmZmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZmZddlmZdd lmZmZdd lmZdd lmZmZdd lmZdd lmZddlmZddl m!Z!ddl"m#Z#ddl$m%Z%d„Z&d„Z'de fd„ƒYZ(de(fd„ƒYZ)dS(iÿÿÿÿ(tprint_functiontdivision(tAdd(t is_sequence(tTuple(tExpr(tMul(tEqualityt Relational(tS(tSymboltDummy(tsympify(tpiecewise_foldt Piecewise(tBooleanFunction(tMatrix(tIdx(tInterval(tflatten(tsiftcO sÄt|ƒ}t|dƒrat|tƒra|j}|j}t||||Ž||||ŽƒS|tjkrwtjS|r’t|Œ\}}nT|j }t |ƒdkrÀt d|ƒ‚ng|D]} t | ƒ^qÇd}}x2|t |ƒkrt|jƒ|}|j}qéWi} tg|D]} | d^q+ƒ} x6|jtƒD]%} | j| ŒsTtƒ| | lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pys As (R thasattrt isinstanceRtlhstrhsR tNaNt_process_limitst free_symbolstlent ValueErrorRttypetlisttlimitstfunctiontsettatomsRthasR txreplaceR titems(tclsR%tsymbolst assumptionsRRR$t orientationtfreetstrepstitsymbols_of_integrationtp((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyt _common_news6    & # cG s¡g}d}xˆ|D]€}t|ttfƒrX|jtƒjƒ}||jƒf}nt|tƒsyt|dtƒr÷t|t ƒrÞ|j d ks¦|j d kr¼|j t|ƒƒqñ|j t||j |j ƒƒq|j t|ƒƒqn†t|tƒr}tt|ƒƒ}t|dtt fƒsGt|ddtƒr}|d}t|ƒdkr—t|dtƒr—|dj|djg|d)nt|ƒdkrø|dd krÙ|dd k rÙ|dg}nm|dd k r|dd kr|d9}|dg}n3|dd kr<|dd kr<g}n |d}|j t||Œƒt|dt ƒr|dj d k r¯t|d|dj kƒ r¯tdƒ‚n|dj d k ròt|d|dj kƒ ròtdƒ‚qòqqqzt|ƒdks,t|ƒdkrE|dd krE|j t|ƒƒqqzt|ƒdkrz|j t||dƒƒqqzq}ntd t|ƒƒ‚qW||fS( sProcess the list of symbols and convert them to canonical limits, storing them as Tuple(symbol, lower, upper). The orientation of the function is also returned when the upper limit is missing so (x, 1, None) becomes (x, None, 1) and the orientation is changed. it _diff_wrtiiiiÿÿÿÿs"Summation exceeds Idx lower range.s"Summation exceeds Idx upper range.sInvalid limits given: %sN(RRRR'R tpoptas_settgetattrtFalseRtlowertNonetuppertappendRRR RR RtstarttendtboolR!tstr(R,R$R.tVtvariablet newsymboltnlim((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyRFsV !"/ %!      114 tExprWithLimitscB s†eZdgZd„Zed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ ed„ƒZ d„Z d „Z RS( tis_commutativec O s©t||||Ž}t|ƒtkr9|\}}}n|Std„|Dƒƒrbtdƒ‚ntj||}|g}|j|ƒt|ƒ|_|j |_ |S(Ncs s-|]#}t|ƒdkp$d|kVqdS(iN(R R<(Rtl((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pys Œss:ExprWithLimits requires values for lower and upper bounds.( R5R"ttupletanyR!Rt__new__textendt_argsRH( R+R%R,R-tpreR$t_tobjtarglist((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyRLƒs   cC s |jdS(s%Return the function applied across limits. Examples ======== >>> from sympy import Integral >>> from sympy.abc import x >>> Integral(x**2, (x,)).function x**2 See Also ======== limits, variables, free_symbols i(RN(tself((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyR%—scC s |jdS(s+Return the limits of expression. Examples ======== >>> from sympy import Integral >>> from sympy.abc import x, i >>> Integral(x**i, (i, 1, 3)).limits ((i, 1, 3),) See Also ======== function, variables, free_symbols i(RN(RS((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyR$ªscC sg|jD]}|d^q S(s^Return 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 transform : Perform mapping on the dummy variable i(R$(RSRI((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyt variables½scC s0g|jD]"}t|ƒdkr |d^q S(s­Return only variables that are dummy variables. Examples ======== >>> from sympy import Integral >>> from sympy.abc import x, i, j, k >>> Integral(x**i, (i, 1, 3), (j, 2), k).bound_symbols [i, j] See Also ======== function, limits, free_symbols as_dummy : Rename dummy variables transform : Perform mapping on the dummy variable ii(R$R (RSRI((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyt bound_symbolsÏscC s£|j|j}}|j}x€|D]x}t|ƒdkrR|j|dƒq#n|d|krv|j|dƒnx"|dD]}|j|jƒqWq#W|S(s4 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} ii(R%R$RR taddtremovetupdate(RSR%R$tisymstxabR2((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyRäs  cC s|j S(s7Return True if the Sum has no free symbols, else False.(R(RS((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyt is_numberscC sTg|jD]+}|d|kr&|n |||f^q }|j}|j||ŒS(Ni(R$R%R(RStxtatbR2R$t integrand((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyt_eval_intervals; c C sÐddlm}m}|jt|jƒ}}|jƒt|tƒ s^|j j |j ƒrÛt }xÉt |ƒD]»\}} dt | ƒkrÃ|| dkrÃ|jr´|f} qÃ||f} nt| dg| dD]} | j||ƒ^qØŒ||>> 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 iÿÿÿÿ(t AppliedUndeftUndefinedFunctioniis.substitution can not create dummy dependenciesii(tsympy.core.functionRaRbR%R#R$treverseRR Rt intersectiontTruet enumerateR R6Rt_subsR:R&RTR'targstissubsetR!tsubstis_zeroR( RStoldtnewRaRbRR$t sub_into_funcR2RZRItsy2tsy1((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyt _eval_subs sD "  :%'!  : ' ( t__name__t __module__t __slots__RLtpropertyR%R$RTRURR[R`Rr(((s>lib/python2.7/site-packages/sympy/concrete/expr_with_limits.pyRG€s   t AddWithLimitscB sDeZdZd„Zd„Zd„Zd„Zd„Zd„ZRS(sZRepresents unevaluated oriented additions. 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