gg*ddlmZddlmZddlmZddlmZddlZddl m Z m Z ddl m Z m Z dd lmZdd lmZmZmZdd lmZmZmZdd lmZdd lmZddlmZddlmZddl m!Z!ddl"m#Z#m$Z$m%Z%ddl&m'Z'm(Z(ddl)m*Z*m+Z+ddl,m-Z-erddl.m/Z/ddl0m1Z1dZ2eGdde eZ3Gdde e3Z4dZ5Gdde3Z6dZ7Gd d!Z8dd"l9m:Z:dd#l;mZ>dd%l?m@Z@mAZAdd&lBmCZCdd'lDmEZEdd(l.mFZFmGZGmHZHmIZImJZJy))) annotations) TYPE_CHECKING)Iterable)reduceN)sympify_sympify)BasicAtom)S) EvalfMixin pure_complexDEFAULT_MAXPREC)call_highest_prioritysympify_method_argssympify_return)cacheit) mod_inverse)default_sort_key) NumberKindsympy_deprecation_warning)as_int func_name filldedent) has_varietysift)mpf_log prec_to_dps) giant_steps)Number) defaultdictcg}g}tj|D]8}|j|}|s|j|(|j|:t|t|fSN)Add make_argscoeffappend)eqccononicis D/var/www/openai/venv/lib/python3.12/site-packages/sympy/core/expr.py_coremr0sZ B C ]]2  WWQZ JJqM IIbM  8S#Y cfeZdZUdZdZded<dZedZe ddZ d Z d Z ed Z ed Zd ZdZddZedgeeddZedgeeddZedgeeddZedgeeddZedgeeddZedgeeddZedgeeddZdddZedgeed d!Zedgeed"d#Zedgeed$d%Zedgeed&d'Z edgeed(d)Z!edgeed*d+Z"edgeed,d-Z#edgeed.d/Z$edgeed0d1Z%d2Z&d3Z'd4Z(edged5Z)edged6Z*edged7Z+edged8Z,d9Z-dfd: Z.e/d;Z0ed<Z1dd=Z2d>Z3dd?Z4d@Z5dAZ6dBZ7dCZ8dDZ9dEZ:dFZ;dGZdJZ?dKZ@eAdLZBddMZCdNZDddOZEdPZFdQZGdRZHdSZIddTZJddUZKddVZLdWZMdXZNddYZOddZZPd[ZQd\ZRdd]ZSdd^ZTdd_ZUd`ZVddaZWdbZXdcZYddZZdeZ[edfZ\dgZ]ddhZ^diZ_djZ`dkZadlZbdmZcdnZddoZedpZfddrZgddsZhdtZidduZjddvZkddwZldxZmddyZnddzZoe ddqd{d|Zpdd}Zqdd~ZrddZsdddZtdddZu ddZvddZwdZxdZye/ddZze ddZ{dZ|ddZ}ddZ~ddZdZddZdZdZdZdZdZdZdZdZdZddZeZdZxZS)Expra Base class for algebraic expressions. Explanation =========== Everything that requires arithmetic operations to be defined should subclass this class, instead of Basic (which should be used only for argument storage and expression manipulation, i.e. pattern matching, substitutions, etc). If you want to override the comparisons of expressions: Should use _eval_is_ge for inequality, or _eval_is_eq, with multiple dispatch. _eval_is_ge return true if x >= y, false if x < y, and None if the two types are not comparable or the comparison is indeterminate See Also ======== sympy.core.basic.Basic ztuple[str, ...] __slots__Tcy)a~Return True if one can differentiate with respect to this object, else False. Explanation =========== Subclasses such as Symbol, Function and Derivative return True to enable derivatives wrt them. The implementation in Derivative separates the Symbol and non-Symbol (_diff_wrt=True) variables and temporarily converts the non-Symbols into Symbols when performing the differentiation. By default, any object deriving from Expr will behave like a scalar with self.diff(self) == 1. If this is not desired then the object must also set `is_scalar = False` or else define an _eval_derivative routine. Note, see the docstring of Derivative for how this should work mathematically. In particular, note that expr.subs(yourclass, Symbol) should be well-defined on a structural level, or this will lead to inconsistent results. Examples ======== >>> from sympy import Expr >>> e = Expr() >>> e._diff_wrt False >>> class MyScalar(Expr): ... _diff_wrt = True ... >>> MyScalar().diff(MyScalar()) 1 >>> class MySymbol(Expr): ... _diff_wrt = True ... is_scalar = False ... >>> MySymbol().diff(MySymbol()) Derivative(MySymbol(), MySymbol()) Fr4selfs r/ _diff_wrtzExpr._diff_wrtGsRr1Nc |j\}}|jrX|jtjur,t d|j tj}}n |j\}}ntj}|jr|jf}n|jr t|f}nn|jr|j|}n+|jr|j!|}n |j}t#|Dcgc]}t%||c}}t'|t#|f}|j|}|j)|||fScc}w)Nexporder) as_coeff_Mulis_Powbaser Exp1Functionr;Oneargsis_Dummysort_keyis_Atomstris_Addas_ordered_termsis_Mulas_ordered_factorstuplerlen class_key)r8r=r'exprr;rDargs r/rFz Expr.sort_keyrs*'') t ;;yyAFF",HUODHH5quuc II c%%C ==MMO%D \\ILue$ $r1__rmod__ct||Sr$Modrms r/__mod__z Expr.__mod__ror1rct||Sr$rrms r/rz Expr.__rmod__rqr1 __rfloordiv__c$ddlm}|||z SNr)floor#sympy.functions.elementary.integersrr8rjrs r/ __floordiv__zExpr.__floordiv__s >TE\""r1rc$ddlm}|||z Srrrs r/rzExpr.__rfloordiv__s >UT\""r1 __rdivmod__c<ddlm}|||z t||fSrrrrrs r/ __divmod__zExpr.__divmod__%s" >TE\"Ce$444r1rc<ddlm}|||z t||fSrrrs r/rzExpr.__rdivmod__+s" >UT\"Ct$444r1c|js td|jd}|js td|tj tj tjfvrtd|zt|}|s|St|ra||kDtjur|S||ktjur|dz S|j|}| td|r|S||dkDrdz Sdz S|S) NzCannot convert symbols to intzCannot convert complex to intzCannot convert %s to intrz#cannot compute int value accuratelyr) is_numberrround is_Numberr NaNInfinityNegativeInfinityint int_valuedtrueequals)r8rr-oks r/__int__z Expr.__int__1s~~;< < JJqM{{;< <  A$6$67 76:; ; FH a=qQVV#qQVV#1u QBz EFFQU+ ++ +r1c|j}|jr t|S|jr|j dr t dt d)NrzCannot convert complex to floatz"Cannot convert expression to float)evalfrfloatr as_real_imagr)r8results r/ __float__zExpr.__float__KsS   =    3 3 5a 8=> ><==r1c|j}|j\}}tt|t|Sr$)rrcomplexr)r8rreims r/ __complex__zExpr.__complex__Vs4$$&BuRy%),,r1c ddlm}|||S)Nr) GreaterThan) relationalr)r8rjrs r/__ge__z Expr.__ge__[s+4''r1c ddlm}|||S)Nr)LessThan)rr)r8rjrs r/__le__z Expr.__le__`s(e$$r1c ddlm}|||S)Nr)StrictGreaterThan)rr)r8rjrs r/__gt__z Expr.__gt__es1 u--r1c ddlm}|||S)Nr)StrictLessThan)rr)r8rjrs r/__lt__z Expr.__lt__js.dE**r1cF|js tdt|S)Nz'Cannot truncate symbols and expressions)rrIntegerr7s r/ __trunc__zExpr.__trunc__os~~EF F4= r1c2|jr|tjd|}|rdt|j d}|j |}|j rtt||S|jr t||St|)|S)Nz\+?\d*\.(\d+)fr) rrmatchrgroupr is_Integerformatis_Floatsuper __format__)r8 format_specmtprecrounded __class__s r/rzExpr.__format__us~ >>+[9B288A;'**T*%%!#g, <<##!';77w!+..r1c*t|dr tj|j|St|drQ|j\}}tj||}tj||t j z}||zStd)N_mpf__mpc_z#expected mpmath number (mpf or mpc))hasattrFloat_newrrr ImaginaryUnitr)xrrrs r/ _from_mpmathzExpr._from_mpmathsv 1g ::aggt, , Q WWFBB%BB%aoo5B7NAB Br1c:td|jDS)a@Returns True if ``self`` has no free symbols and no undefined functions (AppliedUndef, to be precise). It will be faster than ``if not self.free_symbols``, however, since ``is_number`` will fail as soon as it hits a free symbol or undefined function. Examples ======== >>> from sympy import Function, Integral, cos, sin, pi >>> from sympy.abc import x >>> f = Function('f') >>> x.is_number False >>> f(1).is_number False >>> (2*x).is_number False >>> (2 + Integral(2, x)).is_number False >>> (2 + Integral(2, (x, 1, 2))).is_number True Not all numbers are Numbers in the SymPy sense: >>> pi.is_number, pi.is_Number (True, False) If something is a number it should evaluate to a number with real and imaginary parts that are Numbers; the result may not be comparable, however, since the real and/or imaginary part of the result may not have precision. >>> cos(1).is_number and cos(1).is_comparable True >>> z = cos(1)**2 + sin(1)**2 - 1 >>> z.is_number True >>> z.is_comparable False See Also ======== sympy.core.basic.Basic.is_comparable c34K|]}|jywr$)r).0objs r/ z!Expr.is_number..s6IS3==I)allrDr7s r/rzExpr.is_numbersd6DII666r1cx|j}d}|rfddlm}||||f\} } } } tt t ||D cgc]} || | | | dc} } t |jd|}ni}t |jd}t|d sy|jdk(rBtdtD]/}t |j||}|jdk7s/n|jdk7r!| t|d }|j||Sycc} w#ttf$rYywxYw) aReturn self evaluated, if possible, replacing free symbols with random complex values, if necessary. Explanation =========== The random complex value for each free symbol is generated by the random_complex_number routine giving real and imaginary parts in the range given by the re_min, re_max, im_min, and im_max values. The returned value is evaluated to a precision of n (if given) else the maximum of 15 and the precision needed to get more than 1 digit of precision. If the expression could not be evaluated to a number, or could not be evaluated to more than 1 digit of precision, then None is returned. Examples ======== >>> from sympy import sqrt >>> from sympy.abc import x, y >>> x._random() # doctest: +SKIP 0.0392918155679172 + 0.916050214307199*I >>> x._random(2) # doctest: +SKIP -0.77 - 0.87*I >>> (x + y/2)._random(2) # doctest: +SKIP -0.57 + 0.16*I >>> sqrt(2)._random(2) 1.4 See Also ======== sympy.core.random.random_complex_number rr)random_complex_numberTrationalrsubsN_prec) free_symbolssympy.core.randomrdictlistzipabsrrrrrr rmax)r8nre_minim_minre_maxim_maxfreerrar*bdzirepsnmags r/_randomz Expr._randomsOH    ?7JAq!QS%)(+%)r)>aAqSW(X%)(+,-.D 4::ad:34Dtzz!}%DtW% ::? $A74::d:67::?8 ::?ybM::ad:+ +G(+ *    sD"D''D98D9c :d}|jdd}|jry|j}|syt|}|r||zsy|xs|}|}|r|j }|j ry|Dchc]}|j tus|}}d} ||k(ry |jtt|dgt|zd} | tjur|jddddd} | | tjur |jtt|dgt|zd} | tjur|jddddd} | &| tjur| j!| dury|jdd ddd} | &| tjur| j!| dury|j} | 6| tjur$| j!| dury| jr| n| } |D]O} |j#| } |r| j } | dk7s,t%| d s|jd dr| cSy||}|dury|yycc}w#t$rd} YwxYw#t$rd} Y7wxYw) a Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation =========== If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if ``wrt`` is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag ``failing_number`` is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples ======== >>> from sympy import cos, sin, Sum, S, pi >>> from sympy.abc import a, n, x, y >>> x.is_constant() False >>> S(2).is_constant() True >>> Sum(x, (x, 1, 10)).is_constant() True >>> Sum(x, (x, 1, n)).is_constant() False >>> Sum(x, (x, 1, n)).is_constant(y) True >>> Sum(x, (x, 1, n)).is_constant(n) False >>> Sum(x, (x, 1, n)).is_constant(x) True >>> eq = a*cos(x)**2 + a*sin(x)**2 - a >>> eq.is_constant() True >>> eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True >>> (0**x).is_constant() False >>> x.is_constant() False >>> (x**x).is_constant() False >>> one = cos(x)**2 + sin(x)**2 >>> one.is_constant() True >>> ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True cbddlm}d}||D]}|j}|dury|d}|ryy)Nr)denomsFT)sympy.solvers.solversr is_zero) expressionr retNonedenzs r/check_denominator_zerosz1Expr.is_constant..check_denominator_zerosZsA 4Gj)KK99"G * r1simplifyTNr) simultaneousrFror_realfailing_number)getrrsetrr kindrrrrrNr rrZeroDivisionErrorrdiffr)r8wrtflagsrrrrPsym wrt_numberrrrwderivcds r/ is_constantzExpr.is_constantsV 99Z. >>  #h sTzkT ==?D <<&)CScCHH ,BcS C   IId3taST]#;zExpr.equals..saffQiZr1key)rc34K|]}|jywr$)r)rsis r/rzExpr.equals..s;sr}}src38K|]}|jduyw)FN) is_algebraic)rr-s r/rzExpr.equals.."sD1q~~6sc3&K|]}|v ywr$r4)rr2surdss r/rzExpr.equals..%s9SrrU{Ssc3ZK|]"}|z dk(xr|z dk($yw)rNr4rr2r&srs r/rzExpr.equals..)sCE@C" )R0A5 6 (R 0A 5 6@Cs(+c3RK|]}|gk(xr |k( ywr$r4r8s r/rzExpr.equals..-s8#3.1$-R!#5#:#Px|q?P#P.1s$')TN)sympy.simplify.simplifyr&rr r'sympy.polys.polyerrorsr)sympy.polys.numberfieldsr+r factor_termshasr%r as_coeff_mulrNranyr$rr is_comparableatomsr}rDrsortris_realNotImplementedError is_Symbol)r8rjfailing_expressionr'r)r+rfactorsfacfac_zeroconstantndiffr9solmpr&rr6s ` @@@r/rz Expr.equalssT$ @/7? 5= HTE\2DAxxS!##%a( w> $ 3H1166!93G3GQHE JJ/J 0  a%8C8#';s;;$)DDD#(9S99$)E@CEE#'99"#3.1#3 3'+;F+D1B||#  < 'HM KA:RIB+%&9:sT HH>HH1HHH/H"H/H- H*)H*- H;:H;cddlm}ddlm}|jr/ |j d}|yt|dddk(ry|tjury|jd}|jr|tjf}n t|}|y|\}}|jr |jsy|jdk7r-|jdk7rt!| xr|r |dkDS|dkS|jdk(rN|r|jdk(r<|j#r+|j%t&s ||j(ryyyyyyy#t $rYywxYw#|t*f$rYywxYw)Nrr*r(rrrF)r=r+r<r)r _eval_evalfrgetattrr rrrZerorrrbool_eval_is_algebraicr?rBrGrF) r8positiver+r)n2frrr-s r/#_eval_is_extended_positive_negativez(Expr._eval_is_extended_positive_negativeHs_?7 >> %%a( zr7A&!+QUU{ 1 Azz166 $Q}DAqKKAKKww!|1 EH(AIIQIIAqAGGqL++-dhhx6H)$/99$:7I--99   4%&9:s#E 1E EE E'&E'c&|jdS)NTrVrYr7s r/_eval_is_extended_positivezExpr._eval_is_extended_positiveos777FFr1c&|jdS)NFr[r\r7s r/_eval_is_extended_negativezExpr._eval_is_extended_negativers777GGr1c nddlmddlm}ddlmmddlm}ddl m }  tdfd}k(rtjS|d }|tjur|S|d } xr| |z S| |z } jr jrkr |} n |} |j!j#d | } j%|D]} | || j&d| z}  | D]}| tjur| S|js%|k|kcxk(rd k(rnn| |d |dzz } U|k|kcxk(rd k(sgnj| |d|dz z }  | S| S#t($rY| SwxYw)a^ Returns evaluation over an interval. For most functions this is: self.subs(x, b) - self.subs(x, a), possibly using limit() if NaN is returned from subs, or if singularities are found between a and b. If b or a is None, it only evaluates -self.subs(x, a) or self.subs(b, x), respectively. r AccumBoundslog)limitLimit)Interval)solvesetz"Both interval ends cannot be None.cr|rn}|tjSj |}|jtjtj tj tjr>kdk7r ||rdnd}n ||rdnd}t|r td|S)NF+-zCould not compute limit) r rSrr?rrrComplexInfinity isinstancerF) leftr*Crbrfrrrer8rs r/_eval_endpointz+Expr._eval_interval.._eval_endpointsqAyvv IIaO55 A,>,>**K9A%'!$1TcsC!$1TcsC!!U+12KLLHr1T)rnFr)domainrjrk)!sympy.calculus.accumulationboundsrb&sympy.functions.elementary.exponentialrdsympy.series.limitsrerfsympy.sets.setsrgsympy.solvers.solvesetrhrr rSrrBcancelas_numer_denomrCrDr)r8rrrrdrgrhrpABvaluerq singularitieslogtermr9rbrfres```` @@@r/_eval_intervalzExpr._eval_intervalus B>4,3 I!)AB B  " 666M  % :H  & G! q5LA ??q1u!!Q!!Q%T[[]%A%A%CA%FM::c? -a!!1#!# + &A~ ?? A1q51T1%aC"8!85q!S;Q!QQa%QU3t3tQ3!7%aC:P!PP' u    sF'4AF'F'' F43F4cyr$r4rms r/ _eval_powerzExpr._eval_powersr1c>|jr|S|jr| Syr$)is_extended_real is_imaginaryr7s r/_eval_conjugatezExpr._eval_conjugates%  K   5Lr1cddlm}||S)z(Returns the complex conjugate of 'self'.r conjugate)rhrrds r/rzExpr.conjugatesGwr1c4|jrtjSddlm}tj}|}|r|tj z }|j |}|j|d}|tjur|j|d}|tjur7 |j|\}}|j||r t |tjk7rn|r||zzS#t$r|j|d}Y aeeOF((1+CHHQNE~ !Q))),"||AHE1yyQ((l*)T6\!! ",IIaOE,s$5C99DDcddlm}|js |jr|S|jr||S|j r || Sy)Nrr)rhr is_complex is_infinite is_hermitianis_antihermitian)r8rs r/_eval_transposezExpr._eval_transposesGB OOt//K   T? "  " "dO# ##r1cddlm}||S)Nr) transpose)rhr)r8rs r/rzExpr.transposesBr1cddlm}m}|jr|S|jr| S|j }|||S|j }|||Sy)Nr)rr)rhrrrrrr)r8rrrs r/ _eval_adjointzExpr._eval_adjointsbM   K  " "5L""$ ?S> !""$ ?S> ! r1cddlm}||S)Nr)adjoint)rhr)r8rs r/rz Expr.adjoints@t}r1cddlm}tdd}|d}n|d}|rdd|fdfd }||fS) z+Parse and configure the ordering of terms. r) monomial_key startswithNFzrev-c rt|Dcgc]}t|tr|n| c}Scc}wr$)rMrm)monommnegs r/rzExpr._parse_order..negs4%P%QJq%$8#a&qb@%PQ QPs"4c|\}\\}}}} |}t|Dcgc]}|j c}}t||f||ff}|||fScc}w)Nr<)rMrFrT) termrrrrncparter' monom_keyrr=s r/r0zExpr._parse_order..keyst+/ (A(R% %()EVDVAJJUJ3VDEF2h^b"X.E&%' 'EsA)sympy.polys.orderingsrrR)clsr=rrreverser0rrs ` @@r/ _parse_orderzExpr._parse_order sZ 7UL$7  G (Gab  '  R (G|r1c|gS)z4Return list of ordered factors (if Mul) else [self].r4)r8r=s r/rLzExpr.as_ordered_factors(s v r1cpddlm}m}ddlm} ||g|i|}|j sy|S#||f$rYywxYw)aConverts ``self`` to a polynomial or returns ``None``. Explanation =========== >>> from sympy import sin >>> from sympy.abc import x, y >>> print((x**2 + x*y).as_poly()) Poly(x**2 + x*y, x, y, domain='ZZ') >>> print((x**2 + x*y).as_poly(x, y)) Poly(x**2 + x*y, x, y, domain='ZZ') >>> print((x**2 + sin(y)).as_poly(x, y)) None r)PolynomialErrorGeneratorsNeeded)PolyN)r<rrsympy.polys.polytoolsris_Poly)r8gensrDrrrpolys r/as_polyz Expr.as_poly,sM& M. ,t,t,D<< !12  s++55c0ddlmm||jrfd}t t j ||}t|dk(rt|dfrst|dtr`t tj |d|}t|dk(r/t|dr |djr|djr|S|j|\}}|j\}}td|Dst |||} n^gg} } |D]8\} } | js| j!| | f&| j!| | f:t | |d t | |d z} |r| |fS| D cgc]\} }|  c}} Scc}} w) a" Transform an expression to an ordered list of terms. Examples ======== >>> from sympy import sin, cos >>> from sympy.abc import x >>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms() [sin(x)**2*cos(x), sin(x)**2, 1] r)r! NumberSymbolc"t|f Sr$)rm)rr!rs r/r.z'Expr.as_ordered_terms..bsz!fl-CDDr1r/rrc3:K|]\}}|jywr$)is_Order)rrrs r/rz(Expr.as_ordered_terms..qs6WT14==s)r0rT)numbersr!rrIsortedr%r&rNrmr[ is_positive is_negativeras_termsrArr()r8r=datar0add_argsmul_argsrtermsrordered_terms_orderrreprrr!rs @@r/rJzExpr.as_ordered_termsNsx 2 =T[[ECcmmD1s;HH "x{V\,BCx{C0!#-- "<#FMQ&"8A;7  //  //#O((/ Wmmo t666UW=GFF# d}}MM4,/MM4,/ $ Vd;S$78G D= (/0WT1D0 00s Fchddlm}tg}}tj|D]}|j \}}t |}ig}}|tjurutj|D]]} | jr |t | z}| jr"|| \} } | || <|j| M|j| _|j |j"f}t%|}|j||||fft'|t(}t+|i} } t-|D] \}}|| |< g}|D]N\}\}}}dg| z}|j/D] \} } | || | <|j||t%||ffP||fS#ttf$rYwxYw)z,Transform an expression to a list of terms. r)decompose_powerr/r) exprtoolsrrr%r&r>rr rCr[rrrraaddr(realimagrMrrrN enumerateitems)r8rrrrr'_termcpartrfactorr@r;kindicesr-grrs r/rz Expr.as_termss.eReMM$'D,,.LE5ENE6EAEE!!mmE2F''%!WV_4E%,,$3F$; c&)d  f-3"JJ *E6]F LL$v 67 87(:d 01Y7dODAqGAJ$,1 (D(5%CEE"[[] c'*gdm$+ MM4%uv!>? @ -2t|G!*:6! !sFF10F1c|S)z1Removes the additive O(..) symbol if there is oner4r7s r/removeOz Expr.removeOs r1cy)z=Returns the additive O(..) symbol if there is one, else None.Nr4r7s r/getOz Expr.getOsr1c|j}|y|jrC|j}|tjurtj S|j rtjS|jr|jdS|jr|jD]}|j rtjcS|js.ddl m }m }|j|}t|dk(sV|j}|j!||dd}|j"j s|j$j&st)|j$cSt+d|z)a Returns the order of the expression. Explanation =========== The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples ======== >>> from sympy import O >>> from sympy.abc import x >>> (1 + x + O(x**2)).getn() 2 >>> (1 + x).getn() NrDummySymbolrTr[znot sure of order of %s)rrrPr rCrSrGr?rDrKsymbolrrrCrNpoprr@r; is_RationalrrF)r8ooirrsymsrs r/getnz Expr.getns( IIK 9 ZZAAEEzvv {{uu xxvvay xx&&B|| uu yy9!xx/t9> $ A!#E#,E!FB!ww00RVV5G5G'*266{ 2!"";a"?@@r1c ddlm}|||S)Nr) count_ops)functionr)r8visualrs r/rzExpr.count_opss'v&&r1c 8|jrt|j}n|g}t|D]\}}|jr|d|}||d}n|}g}|rN|rL|dj r=|dj r.|dtjurtj|d g|dd|rlt|} t|}| rT|rRt|| k7rDtd|D cgc],} t|jj| dkDs+| .c} z||gScc} w)aReturn [commutative factors, non-commutative factors] of self. Explanation =========== self is treated as a Mul and the ordering of the factors is maintained. If ``cset`` is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting ``warn`` to False. Note: -1 is always separated from a Number unless split_1 is False. Examples ======== >>> from sympy import symbols, oo >>> A, B = symbols('A B', commutative=0) >>> x, y = symbols('x y') >>> (-2*x*y).args_cnc() [[-1, 2, x, y], []] >>> (-2.5*x).args_cnc() [[-1, 2.5, x], []] >>> (-2*x*A*B*y).args_cnc() [[-1, 2, x, y], [A, B]] >>> (-2*x*A*B*y).args_cnc(split_1=False) [[-2, x, y], [A, B]] >>> (-2*x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: >>> (-2 + x + A).args_cnc() [[], [x - 2 + A]] >>> (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] Nrrz"repeated commutative arguments: %s) rKrrDrraris_extended_negativer rcrNrrcount) r8csetwarnsplit_1rDr-mir*ncclenr.s r/args_cncz Expr.args_cncsN ;; ?D6Dt_EAr$$!H!"X % AB  aDNN aD % %!AMM)]]QqTE*AbqE q6DAAQ4 !E/0!RqDO4I4I"4MPQ4Q"q!R"STT2w"Ss ,D D c t|}t|tstjSt |}|stjS||k(r%|dk(rtj StjS|tj ur`tj|Dcgc]'}|jdtj ur|)c}stjStS|dk(r|jr|jr|j||}|stjS|s/|ttj|Dcgc]}||z c}z S|ttj|Dcgc]}||z c}z S|j|ddS||z}d}dd}gtj|} |j} |j} | r| stjS|r|jr| s| stj|} ttj|j|dD cgc] t fd| Dr c} }|j||d }|jr|s|St!|j"D cgc]} | j%c} \}}t'|r|St|D cgc]} tj(| c} S| xs| }|j%dt+|  \} }| D]}|j%dt+|  \}}|g}t-| t-|kDr?|j/| }t-|t-| zt-|k(st|r$j1tt3||zj1||f|r gk(rtjSrtSy stjStfd Drk|dd||}|Y|s:ttDcgc] \}}t|c}}tddd |Stdd|t-|zd St5|dD}|r||||}||sTdd}t7dt-D] |j9 d}|rntt3||d |zS|t-|z}tDcgc]\}}tt3|||d zc}}St3t;t5|dD}|rb||||}|V|s=tDcgc])\}}tt3||d t-| |zz+c}}St||t-|zd Sd }t=D]-\ \}}||||}||s|||f}tjS|r6|\}}}|stt3||d |zSt||t-|zd StjScc}wcc}wcc}wcc} wcc} wcc} wcc}}wcc}}wcc}}w)a Returns the coefficient from the term(s) containing ``x**n``. If ``n`` is zero then all terms independent of ``x`` will be returned. Explanation =========== When ``x`` is noncommutative, the coefficient to the left (default) or right of ``x`` can be returned. The keyword 'right' is ignored when ``x`` is commutative. Examples ======== >>> from sympy import symbols >>> from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: >>> (-x + 2*y).coeff(-1) x >>> (x - 2*y).coeff(-1) 2*y You can select terms with no Rational coefficient: >>> (x + 2*y).coeff(1) x >>> (3 + 2*x + 4*x**2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): >>> (3 + 2*x + 4*x**2).coeff(x, 0) 3 >>> eq = ((x + 1)**3).expand() + 1 >>> eq x**3 + 3*x**2 + 3*x + 2 >>> [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] >>> eq -= 2 >>> [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: >>> (-x - 2*y).coeff(2) -y >>> from sympy import sqrt >>> (x + sqrt(2)*x).coeff(sqrt(2)) x The matching is exact: >>> (3 + 2*x + 4*x**2).coeff(x) 2 >>> (3 + 2*x + 4*x**2).coeff(x**2) 4 >>> (3 + 2*x + 4*x**2).coeff(x**3) 0 >>> (z*(x + y)**2).coeff((x + y)**2) z >>> (z*(x + y)**2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: >>> (x + z*(x + x*y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: >>> from sympy import factor_terms >>> factor_terms(x + z*(x + x*y)).coeff(x) z*(y + 1) + 1 >>> n, m, o = symbols('n m o', commutative=False) >>> n.coeff(n) 1 >>> (3*n).coeff(n) 3 >>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m 1 + m >>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m m If there is more than one possible coefficient 0 is returned: >>> (n*m + m*n).coeff(n) 0 If there is only one possible coefficient, it is returned: >>> (n*m + x*m*n).coeff(m*n) x >>> (n*m + x*m*n).coeff(m*n, right=1) 1 See Also ======== as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used rr)rightT)as_Addc|r|sgStt|t|}t|D]}||||k7s|d|cS|ddSr$)minrNrange)l1l2rr-s r/incommonzExpr.coeff..incommonsSR CGSW%A1Xa5BqE>bq6Ma5Lr1crrttkDryt}|s jjtt|z dzD]#tfdt|Ds#nd|s jj|st|zz S)a Find where list sub appears in list l. When ``first`` is True the first occurrence from the left is returned, else the last occurrence is returned. Return None if sub is not in l. Examples ======== >> l = range(5)*2 >> find(l, [2, 3]) 2 >> find(l, [2, 3], first=0) 7 >> find(l, [2, 4]) None Nrc3:K|]}|z|k(ywr$r4)rjr-lsubs r/rz+Expr.coeff..find..s#<8aqQx3q6)8)rNrrr)rrfirstrr-s`` @r/findzExpr.coeff..finds"a3s8c!f#4CA  3q6A:>*<58<<+  }UFa!e$Hr1c3LK|]}|tjvywr$)r[r&)rxir-s r/rzExpr.coeff..s >"rS]]1--s!$F)r_first)rrNc3:K|]\}}|ddk(yw)rrNr4)rrrr+s r/rzExpr.coeff..s"0RTQ11a=Rrc3&K|] }|d ywrNr4rrs r/rzExpr.coeff..&s#5"QAaD"sc3JK|]}tt|dywr)rreversedr s r/rzExpr.coeff..5s!C1$x!~"6s!#T)rrmr r rSrrCr%r&r>rIr'as_independentrar[rrrDrrrbrTrN differencer(rrr intersectionr r)r8rrrrrr*rrrDself_cx_cxargsr-rrvc_partnc_partone_cnxmargsrresidiirbeggcdcrendhitr+s ` @r/r'z Expr.coeff6s:` AJ!U#66M 1I66M 9Avuu 66M : ]]4030^^%a(AEE103Bvv 8O 6xxDKKJJqJ.66M#S]]15E'F5E!5E'F"GGGcq1A#B1AAAaC1A#BCCC&&q&6q9 9 qD ! F}}T"$$ #66M dkk&MM!$Et/B/B1/Ea/H!I@!IA>>>!I@AA%6B99E !"''#B'QAJJL'#BCOFG6" G "3R(@RTQaR(@#A31aQTRT CVWW"BqE!HR#b'\]$;<<#5"#56C#r5)> !!uQx!&q#b'!2A#'#4#4RU1X#>D#' %"3 #T$Z#cr(%:<<RL"$LAS47QqrU?%<$LMMxx!C!CDFGC#r5)> "VX$YVXdaQRS47QS B5G+G%IVX$YZZ"CSW $677C&r] 6Aq!R'> !Qh66M+"HB1 "T!Wq"v%577"Ab3r7lm$45566Ms3(G#Bv@ $C=<)A %M%Zs6,X> Y 7 Y Y YY3Y !Y" /.Y( c|S)a/ Convert a polynomial to a SymPy expression. Examples ======== >>> from sympy import sin >>> from sympy.abc import x, y >>> f = (x**2 + x*y).as_poly(x, y) >>> f.as_expr() x**2 + x*y >>> sin(x).as_expr() sin(x) r4)r8rs r/as_exprz Expr.as_exprPs $ r1cR|j|}|r|j|s|Syy)a, Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples ======== >>> from sympy import E, pi, sin, I, Poly >>> from sympy.abc import x >>> E.as_coefficient(E) 1 >>> (2*E).as_coefficient(E) 2 >>> (2*sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) >>> (2*E + x*E).as_coefficient(E) x + 2 >>> _.args[0] # just want the exact match 2 >>> p = Poly(2*E + x*E); p Poly(x*E + 2*E, x, E, domain='ZZ') >>> p.coeff_monomial(E) 2 >>> p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient ``2*x`` is desired then the ``coeff`` method should be used.) >>> (2*E*x + x).as_coefficient(E) >>> (2*E*x + x).coeff(E) 2*x >>> (E*(x + 1) + x).as_coefficient(E) >>> (2*pi*I).as_coefficient(pi*I) 2 >>> (2*I).as_coefficient(pi*I) See Also ======== coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used N)extract_multiplicativelyr?)r8rPrs r/as_coefficientzExpr.as_coefficientds.@  ) )$ / QUU4[H!1r1c:ddlm}ddlm}ddlm}|t jur||fS|j}|jdt|trt}nt}tg|D]1}t||rj|!j|3fd} ||us|tur,|tur$| |r|j|fS||jfS|turt!|j"} n|j%\} } t'| | }|d} |d} |tur t| || fSt) D]4\}}| |r| j+| |d n| j|6t| | r t| d difS|| fS) a A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: * separatevars() to change Mul, Add and Pow (including exp) into Mul * .expand(mul=True) to change Add or Mul into Add * .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for `self` of zero regardless of hints. For nonzero `self`, the returned tuple (i, d) has the following interpretation: * i will has no variable that appears in deps * d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) * if self is an Add then self = i + d * if self is a Mul then self = i*d * otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples ======== -- self is an Add >>> from sympy import sin, cos, exp >>> from sympy.abc import x, y, z >>> (x + x*y).as_independent(x) (0, x*y + x) >>> (x + x*y).as_independent(y) (x, x*y) >>> (2*x*sin(x) + y + x + z).as_independent(x) (y + z, 2*x*sin(x) + x) >>> (2*x*sin(x) + y + x + z).as_independent(x, y) (z, 2*x*sin(x) + x + y) -- self is a Mul >>> (x*sin(x)*cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul >>> from sympy import symbols >>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False) >>> (n1 + n1*n2).as_independent(n2) (n1, n1*n2) >>> (n2*n1 + n1*n2).as_independent(n2) (0, n1*n2 + n2*n1) >>> (n1*n2*n3).as_independent(n1) (1, n1*n2*n3) >>> (n1*n2*n3).as_independent(n2) (n1, n2*n3) >>> ((x-n1)*(x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: >>> (sin(x)).as_independent(x) (1, sin(x)) >>> (sin(x)).as_independent(y) (sin(x), 1) >>> exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: >>> (3*x).as_independent(x, as_Add=True) (0, 3*x) -- force self to be treated as a Mul: >>> (3+x).as_independent(x, as_Add=False) (1, x + 3) >>> (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. >>> (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols >>> from sympy import Integral >>> I = Integral(x, (x, 1, 2)) >>> I.has(x) True >>> x in I.free_symbols False >>> I.as_independent(x) == (I, 1) True >>> (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values >>> from sympy import separatevars, log >>> separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) >>> (x + x*y).as_independent(y) (x, x*y) >>> separatevars(x + x*y).as_independent(y) (x, y + 1) >>> (x*(1 + y)).as_independent(y) (x, y + 1) >>> (x*(1 + y)).expand(mul=True).as_independent(y) (x, x*y) >>> a, b=symbols('a b', positive=True) >>> (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also ======== separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul r)r_unevaluated_Add_unevaluated_Mulrcj|j}s|S|xs|j|jzS)zreturn the standard has() if there are no literal symbols, else check to see that symbol-deps are in the free symbols.)r?r)r has_otherrjrs r/r?z Expr.as_independent..hasEs<u I  >(< > >r1TFNevaluate)rrrr&mulr(r rSfuncrrmr%r[rr(identityrrDrrrextend)r8depshintrr&r(r-wantrr?rDrdependindepr-rrjrs @@r/r zExpr.as_independentsL #)) 166>$< yy 88Hjs3 5DDeA!V$  Q   ?  CDO4y t,,dmm,,s{DII==?b sO4% 3;K!16!:; ;!" 1q6MM"QR&) Q & ;02V,e,+ + &)+ +r1c \|jd|k(ryddlm}m}||||fS)aPerforms complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. >>> from sympy import symbols, I >>> x, y = symbols('x,y', real=True) >>> (x + y*I).as_real_imag() (x, y) >>> from sympy.abc import z, w >>> (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) ignoreNr)rr)rrhrr)r8deephintsrrs r/rzExpr.as_real_imaghs.. 99X $ & CtHbh' 'r1cdtt}|j|jg|S)aReturn self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also ======== as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. )r"rupdate as_base_exp)r8rs r/as_powers_dictzExpr.as_powers_dicts+   $""$%&r1cztt}|s}tj|D])}|j \}}||j |+|j D]'\}}t|dk(r |d||<t|||<)n|j|ddi\}} tj| D]} | jr[| j|\}} |tjur|| j |H|| j| j |j| sm|| j tj|Dcic]}|t||}}|tjur!|jtj|itt } | j|| Scc}w)aOReturn a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols ``syms`` are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned. Examples ======== >>> from sympy.abc import a, x, y >>> (3*x + a*x + 4).as_coefficients_dict() {1: 4, x: 3, a*x: 1} >>> _[a] 0 >>> (3*a*x).as_coefficients_dict() {a*x: 3} >>> (3*a*x).as_coefficients_dict(x) {x: 3*a} >>> (3*a*x).as_coefficients_dict(y) {1: 3*a*x} rrrT)r"rr%r&r>r(rrNr rKr@r rC _new_rawargsrSr:r) r8rrair*rrvinddepr-rdis r/as_coefficients_dictzExpr.as_coefficients_dictsy4  mmD)(1! A* 1q6Q;Q4AaD7AaD " +t**D>>HC]]3'88)1>>40DAqAEEz! A.!..!,-44Q7aDKK&()**1C1JA*!&& !%%&   !  +sF8c&|tjfSr$r rCr7s r/r;zExpr.as_base_expsQUU{r1cR|r|j|s|dfStj|ffS)aReturn the tuple (c, args) where self is written as a Mul, ``m``. c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. - if you know self is a Mul and want only the head, use self.args[0]; - if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; - if you want to split self into an independent and dependent parts use ``self.as_independent(*deps)`` >>> from sympy import S >>> from sympy.abc import x, y >>> (S(3)).as_coeff_mul() (3, ()) >>> (3*x*y).as_coeff_mul() (3, (x, y)) >>> (3*x*y).as_coeff_mul(x) (3*y, (x,)) >>> (3*y).as_coeff_mul(x) (3*y, ()) r4)r?r rC)r8r0kwargss r/r@zExpr.as_coeff_muls-< 488T?Rxuutg~r1cR|r|j|s|dfStj|ffS)aReturn the tuple (c, args) where self is written as an Add, ``a``. c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of ``a``; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. - if you know self is an Add and want only the head, use self.args[0]; - if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. - if you want to split self into an independent and dependent parts use ``self.as_independent(*deps)`` >>> from sympy import S >>> from sympy.abc import x, y >>> (S(3)).as_coeff_add() (3, ()) >>> (3 + x).as_coeff_add() (3, (x,)) >>> (3 + x + y).as_coeff_add(x) (y + 3, (x,)) >>> (3 + y).as_coeff_add(x) (y + 3, ()) r4)has_freer rS)r8r0s r/ as_coeff_addzExpr.as_coeff_adds.>  4==$'Rxvvwr1c|s tjtjfS|jd\}}|jr| | }}||fS)aRReturn the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples ======== >>> from sympy.abc import x >>> (3*(x + 1)**2).primitive() (3, (x + 1)**2) >>> a = (6*x + 2); a.primitive() (2, 3*x + 1) >>> b = (x/2 + 3); b.primitive() (1/2, x + 6) >>> (a*b).primitive() == (1, a*b) True Tr)r rCrSr>r)r8r*rs r/ primitivezExpr.primitivesM&55!&&=   $ /1 ==2rqA!t r1c&tj|fS)aThis method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and ``Mul(*foo.as_content_primitive()) == foo``. The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples ======== >>> from sympy import sqrt >>> from sympy.abc import x, y, z >>> eq = 2 + 2*x + 2*y*(3 + 3*y) The as_content_primitive function is recursive and retains structure: >>> eq.as_content_primitive() (2, x + 3*y*(y + 1) + 1) Integer powers will have Rationals extracted from the base: >>> ((2 + 6*x)**2).as_content_primitive() (4, (3*x + 1)**2) >>> ((2 + 6*x)**(2*y)).as_content_primitive() (1, (2*(3*x + 1))**(2*y)) Terms may end up joining once their as_content_primitives are added: >>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() (11, x*(y + 1)) >>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() (9, x*(y + 1)) >>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive() (1, 6.0*x*(y + 1) + 3*z*(y + 1)) >>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive() (121, x**2*(y + 1)**2) >>> ((x*(1 + y) + 0.4*x*(3 + 3*y))**2).as_content_primitive() (1, 4.84*x**2*(y + 1)**2) Radical content can also be factored out of the primitive: >>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2)*(1 + 2*sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. >>> (x/2 + y).as_content_primitive() (1/2, x + 2*y) >>> (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) rF)r8r,clears r/as_content_primitivezExpr.as_content_primitive3snuud{r1c&|tjfS)a/Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also ======== normal: return ``a/b`` instead of ``(a, b)`` rFr7s r/rxzExpr.as_numer_denomlsQUU{r1cddlm}|j\}}|tjur|S|j r ||d|z S||z S)zReturn the expression as a fraction. expression -> a/b See Also ======== as_numer_denom: return ``(a, b)`` instead of ``a/b`` rr')r,r(rxr rCr)r8r(rrs r/normalz Expr.normal|sJ *""$1 :H ;;#Aqs+ +Q3Jr1c ddlm}ddlm}t |}|t j ury|t jur|S||k(rt jS|jr3|j\}}|t jurt||d}|jr9|j\}}|j|}||j|S|S||z } |jrG|t jur|j r't jS|t j"ur8|j$rt jS|j rt j"S|t j&ur|j(st j&S|j*r(| j*sy|j r | j$ry| S|j,r(| j,sy|j r | j$ry| S|j.r(| j.sy|j r | j$ry| Syyyy|j0s|j2s|t j4ur| jr^t7| j8dk(rF| j8dj*rK| j8dj r1| j8d|k(r| S| j*r|jr| Syyyyy|jr|j\} } |jr]|t j:urK| t jur8|j$r| j|  } n| j|} | | | zSy|| k(r| Sg} | j8D](}|j|}|y| j=|*| t jur| Dcgc]}| |z }}||St?j@| S|jrKtC|j8}tE|D]'\}}|j|}||||<t|cSy|jFs tI||r{|jK\}}|jK\}}||k(r |jM|}| tO||Sy||k(r*|jjMd}| tO||Syyycc}w) aReturn None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples ======== >>> from sympy import symbols, Rational >>> x, y = symbols('x,y', real=True) >>> ((x*y)**3).extract_multiplicatively(x**2 * y) x*y**2 >>> ((x*y)**3).extract_multiplicatively(x**4 * y) >>> (2*x).extract_multiplicatively(2) x >>> (2*x).extract_multiplicatively(3) >>> (Rational(1, 2)*x).extract_multiplicatively(3) x/6 r)r;rr%NF)r+r)(rsr;rr&rr rrCrIrMr[rK as_two_termsr"rrrrrrlr rrris_NumberSymbolrGrrNrDrcr(r%rbrrr?rmr;extract_additivelyr})r8r*r;r&ccpcrrrquotientcspsxcnewargsrQnewargtrDr-sbsecbcenew_exps r/r"zExpr.extract_multiplicativelysM4 ?) AJ 155= :K $Y55L 88[[]FBB/ 88>>#DAq--a0A}11!44!8 >>qzz!==::%+++==::%]]---***yy,,,**%%(*>*>#O!!++%%(*>*>#O((%%(*>*>#O !!# !> ! !T^^tq7N3x}}#5#:==#..8==3C3O3OT\TaTabcTdhlTl#O$$*5$Um3O.[[^^%FB{{q 5QUU?}}!::A2>?88;~!"u Bw Gww55a8>v&  &-.g1g.'..~~g.. [[ ?D#D/355a8%$DG:% * [[JtS1%%'FB]]_FBRx//3&r7++'b((55a8&r7++'2/s Sct|}|tjury|jr|S||k(rtjS|tjk(ry|j r5|j sy|}||z }|dkDr |dk\r||ks|dkr |dkr||kDr|Sy|j r,|j \}}|j|}|y||zS|jr|jdj r|j|}|jdxsd|z}|r$|||zz }||j|xs|zxsdS|j \}}|j \}} |j|}|y| j|} | y|| zSt||\}}|jdxsd|z}|r||j|xs|zxsdSg} tj|D]q} | j\} }|j|}|sy|j \}}|j| }|y|||zz}| j||z|zs| j|t| S)apReturn self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples ======== >>> from sympy.abc import x, y >>> e = 2*x + 3 >>> e.extract_additively(x + 1) x + 2 >>> e.extract_additively(3*x) >>> e.extract_additively(4) >>> (y*(x + 1)).extract_additively(x + 1) >>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1) (x + 1)*(x + 2*y) + 3 See Also ======== extract_multiplicatively coeff as_coefficient Nrr)rr rr rSr as_coeff_AddrWrIrDr'r0r%r&r>r()r8r*r+rr`xaxa0hshstxa2coeffsracatcoccots r/rWzExpr.extract_additively s2 AJ 155= 99K $Y66M QVV^ >>;;B6DQ419FtqyTBY  ;;%%'EB&&q)Bz6M 88q ++AB((+0q!3Cbd{t66q9ATBKtK>>#DAq&&(FB&&q)Bz''*C{8O$?D$$Q',1a/ 42215=>G4 Gq!A^^%FBBB(HC''+Bz BrEMD MM38R- ("  dF|r1ctddd|jDchc]}|jD]}|c}}Scc}}w)a0 Like ``free_symbols``, but returns the free symbols only if they are contained in an expression node. Examples ======== >>> from sympy.abc import x, y >>> (x + y).expr_free_symbols # doctest: +SKIP {x, y} If the expression is contained in a non-expression object, do not return the free symbols. Compare: >>> from sympy import Tuple >>> t = Tuple(x + y) >>> t.expr_free_symbols # doctest: +SKIP set() >>> t.free_symbols {x, y} The expr_free_symbols property is deprecated. Use free_symbols to get the free symbols of an expression. 1.9deprecated-expr-free-symbolsdeprecated_since_versionactive_deprecations_target)rrDexpr_free_symbols)r8r-rs r/rzzExpr.expr_free_symbolsw sG. "# &+'E  G  99B9aa.A.A.A9BBBs<cy)a Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False. Examples ======== >>> from sympy.abc import x, y >>> e = x - y >>> {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the ``y - x`` is considered like ``-(x - y)``, since it is in a product without a leading factor of -1, the result is false below: >>> (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use `signsimp`: >>> from sympy import signsimp >>> signsimp(x*(y - x)) -x*(x - y) >>> _.could_extract_minus_sign() True Fr4r7s r/could_extract_minus_signzExpr.could_extract_minus_sign s8r1cVddlm}ddlm}tj }tj }tj|}g}|D]$}t||r||jgz } ||z}&tj } g} tjtjz} |rZ|j} | jr|| jz }.| j r| j#| } | | | z } S| | gz } |rZ| j$r| } d}n| j&| j(\} }|| dz tj*z dz}||dz z }| |z }|r(|j-d}||tj*z }|}| t/|f|zzt/| z}|dk7r |||z}||fS)ac Try to write self as ``exp_polar(2*pi*I*n)*z`` in a nice way. Return (z, n). >>> from sympy import exp_polar, I, pi >>> from sympy.abc import x, y >>> exp_polar(I*pi).extract_branch_factor() (exp_polar(I*pi), 0) >>> exp_polar(2*I*pi).extract_branch_factor() (1, 1) >>> exp_polar(-pi*I).extract_branch_factor() (exp_polar(I*pi), -1) >>> exp_polar(3*pi*I + x).extract_branch_factor() (exp_polar(x + I*pi), 1) >>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor() (y*exp_polar(2*pi*x), -1) >>> exp_polar(-I*pi/2).extract_branch_factor() (exp_polar(-I*pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): >>> exp_polar(I*pi).extract_branch_factor(allow_half=True) (1, 1/2) >>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True) (1, 1) >>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True) (1, 3/2) >>> exp_polar(-I*pi).extract_branch_factor(allow_half=True) (1, -1/2) r) exp_polarceilingr4rr)rsr~rrr rSrCr[r&rmr;PirrrIrDrKr#rrKrHalfrWr%)r8 allow_halfr~rrresrDexpsrQpiimultextrasipir;r'tail branchfactr*rnewexps r/extract_branch_factorzExpr.extract_branch_factor s> E? FFee}}T"C#y) !s   &&dd1??"((*Czz zz**3/$u$G seOF   ED.'..0D0DEKE4U1Wqvv-.q0  Z\ J  %%a(B~QVV SA54<))CL8 Q; 9V$ $CAv r1cz|rttt|}n|j}|sy|j |S)a Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are "polynomials" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, \*syms) should work if and only if expr.is_polynomial(\*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples ======== >>> from sympy import Symbol, Function >>> x = Symbol('x') >>> ((x**2 + 1)**4).is_polynomial(x) True >>> ((x**2 + 1)**4).is_polynomial() True >>> (2**x + 1).is_polynomial(x) False >>> (2**x + 1).is_polynomial(2**x) True >>> f = Function('f') >>> (f(x) + 1).is_polynomial(x) False >>> (f(x) + 1).is_polynomial(f(x)) True >>> (1/f(x) + 1).is_polynomial(f(x)) False >>> n = Symbol('n', nonnegative=True, integer=True) >>> (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. >>> from sympy import sqrt, factor, cancel >>> y = Symbol('y', positive=True) >>> a = sqrt(y**2 + 2*y + 1) >>> a.is_polynomial(y) False >>> factor(a) y + 1 >>> factor(a).is_polynomial(y) True >>> b = (y**2 + 2*y + 1)/(y + 1) >>> b.is_polynomial(y) False >>> cancel(b) y + 1 >>> cancel(b).is_polynomial(y) True See also .is_rational_function() T)rmaprr_eval_is_polynomialr8rs r/ is_polynomialzExpr.is_polynomial s;B s7D)*D$$D''--r1c.||vry|j|syyNTrJrs r/rzExpr._eval_is_polynomialK s" 4<t}}d#$r1c|rttt|}n|j}|s|tvS|j |S)a  Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are "rational functions" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples ======== >>> from sympy import Symbol, sin >>> from sympy.abc import x, y >>> (x/y).is_rational_function() True >>> (x**2).is_rational_function() True >>> (x/sin(y)).is_rational_function(y) False >>> n = Symbol('n', integer=True) >>> (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. >>> from sympy import sqrt, factor >>> y = Symbol('y', positive=True) >>> a = sqrt(y**2 + 2*y + 1)/y >>> a.is_rational_function(y) False >>> factor(a) (y + 1)/y >>> factor(a).is_rational_function(y) True See also is_algebraic_expr(). )rrrr_illegal_eval_is_rational_functionrs r/is_rational_functionzExpr.is_rational_functionS sCf s7D)*D$$D8++..t44r1c2||vry|j|syyr) has_xfreers r/rzExpr._eval_is_rational_function s 4<~~d#$r1c|jstdj|t|}|j ||S)a This tests whether an expression is meromorphic as a function of the given symbol ``x`` at the point ``a``. This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples ======== >>> from sympy import zoo, log, sin, sqrt >>> from sympy.abc import x >>> f = 1/x**2 + 1 - 2*x**3 >>> f.is_meromorphic(x, 0) True >>> f.is_meromorphic(x, 1) True >>> f.is_meromorphic(x, zoo) True >>> g = x**log(3) >>> g.is_meromorphic(x, 0) False >>> g.is_meromorphic(x, 1) True >>> g.is_meromorphic(x, zoo) False >>> h = sin(1/x)*x**2 >>> h.is_meromorphic(x, 0) False >>> h.is_meromorphic(x, 1) True >>> h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. >>> log(x).is_meromorphic(x, -1) True >>> log(x).is_meromorphic(x, 0) False >>> sqrt(x).is_meromorphic(x, -1) True >>> sqrt(x).is_meromorphic(x, 0) False z{} should be of symbol type) is_symbolrrr_eval_is_meromorphicr8rrs r/is_meromorphiczExpr.is_meromorphic s?n{{9@@CD D AJ((A..r1c4||k(ry|j|syyrrrs r/rzExpr._eval_is_meromorphic s 19}}Q r1cz|rttt|}n|j}|sy|j |S)a  This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are "algebraic expressions" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples ======== >>> from sympy import Symbol, sqrt >>> x = Symbol('x', real=True) >>> sqrt(1 + x).is_rational_function() False >>> sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. >>> from sympy import exp, factor >>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1) >>> a.is_algebraic_expr(x) False >>> factor(a).is_algebraic_expr() True See Also ======== is_rational_function References ========== .. [1] https://en.wikipedia.org/wiki/Algebraic_expression T)rrrr_eval_is_algebraic_exprrs r/is_algebraic_exprzExpr.is_algebraic_expr s;V s7D)*D$$D++D11r1c.||vry|j|syyrrrs r/rzExpr._eval_is_algebraic_expr s" 4<t}}d#$r1rc  9|j}|s|St|dkDr td|jddlm}m} t| r|jv} n&|} | |j| ijv} | s| d|fDS|St|dk7s|dvr tdttddl m } m } js4jrjrd nd }n| jrd nd }nr#j r| j#nat%|d k(rt&j(nBt%|d k(rt&j*n#t&j,k(rt&j(t/z r^j rRdd lm} |j5z j7|d d }| fd |DS|j5z Ssdk7rY|j5zzij7d|d |d}| fd|DS|j5z z z iSjj:cxurnj<durQ|dd|j5j7|||}| fd|DS|j5Sddlm ||jC||}|jExst&j,}|rL|jG}||kDr5|dk7r#||j5tI||zz }n|dz }n||krddl%m&}tOddD]}|jC||z|}|jG}||k7s1||||z |z||z z z}|jC||}|jG|kr.|jC||}|dz }|jG|kr.ntdt%|d|||zz }|jE}|jQ}nV|jSr|S|z}|jU} ||zjQ|k(rt&j,} ddl,m-}|||zSfd}||jQj]|S#|$r>|j5zj9|}|j5zcYSwxYw#tV$r|cYSwxYw#tV$r||zcYSwxYw)a Series expansion of "self" around ``x = x0`` yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of "self" around the point ``x = x0`` with respect to ``x`` up to ``O((x - x0)**n, x, x0)`` (default n is 6). If ``x=None`` and ``self`` is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters ========== expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which ``x`` is calculated. Can be any value from ``-oo`` to ``oo``. n : Value The value used to represent the order in terms of ``x**n``, up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If ``dir="+"``, then (x->x0+). If ``dir="-", then (x->x0-). For infinite ``x0`` (``oo`` or ``-oo``), the ``dir`` argument is determined from the direction of the infinity (i.e., ``dir="-"`` for ``oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples ======== >>> from sympy import cos, exp, tan >>> from sympy.abc import x, y >>> cos(x).series() 1 - x**2/2 + x**4/24 + O(x**6) >>> cos(x).series(n=4) 1 - x**2/2 + O(x**4) >>> cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1)) >>> e = cos(x + exp(y)) >>> e.series(y, n=2) cos(x + 1) - y*sin(x + 1) + O(y**2) >>> e.series(x, n=2) cos(exp(y)) - x*sin(exp(y)) + O(x**2) If ``n=None`` then a generator of the series terms will be returned. >>> term=cos(x).series(n=None) >>> [next(term) for i in range(2)] [1, -x**2/2] For ``dir=+`` (default) the series is calculated from the right and for ``dir=-`` the series from the left. For smooth functions this flag will not alter the results. >>> abs(x).series(dir="+") x >>> abs(x).series(dir="-") -x >>> f = tan(x) >>> f.series(x, 2, 6, "+") tan(2) + (1 + tan(2)**2)*(x - 2) + (x - 2)**2*(tan(2)**3 + tan(2)) + (x - 2)**3*(1/3 + 4*tan(2)**2/3 + tan(2)**4) + (x - 2)**4*(tan(2)**5 + 5*tan(2)**3/3 + 2*tan(2)/3) + (x - 2)**5*(2/15 + 17*tan(2)**2/15 + 2*tan(2)**4 + tan(2)**6) + O((x - 2)**6, (x, 2)) >>> f.series(x, 2, 3, "-") tan(2) + (2 - x)*(-tan(2)**2 - 1) + (2 - x)**2*(tan(2)**3 + tan(2)) + O((x - 2)**3, (x, 2)) For rational expressions this method may return original expression without the Order term. >>> (1/x).series(x, n=8) 1/x Returns ======= Expr : Expression Series expansion of the expression about x0 Raises ====== TypeError If "n" and "x0" are infinity objects PoleError If "x0" is an infinity object rz+x must be given for multivariate functions.rc3 K|]}|ywr$r4)rr9s r/rzExpr.series.. s*6a6s z+-zDir must be '+' or '-'r)rsignrjrk PoleError)rrrc3HK|]}|jz ywr$r)rr2rrs r/rzExpr.series.. s!;2BGGAtAv.s")r)x0rrlogxrc3VK|] }|jz z z i"ywr$r)rr2rrrs r/rzExpr.series.. s-C2AdFRW$4 56s&)Trr[rrc3BK|]}|jywr$r)rr9rxposs r/rzExpr.series.. s4AtQsOrderrrrr zCould not calculate z terms for collectc3PK|D]}|js|d}|z}d}t|j} ||z |zj}|z}|r |jr+|jr|t|jz }n|dz }|||k(r||z }dyw)z&Return terms of lseries one at a time.rrN)rIrNrDrr) r9r2yieldedrndidndodorrs r/ yield_lseriesz"Expr.series..yield_lseries sB99   Gb! QADbgg,C 7lQ.779Q!R[[$99 CL0D AID 3;!2 sB#B&)/rrNrrrrrrmxreplacerrhrrr rErrrrHr rCrcrSrrrrseriesaseriesrrGsympy.series.orderr _eval_nseriesrrRationalrrrrr?doitrFsympy.simplify.radsimpr _eval_lseries)r8rrrrrrrrrrBrrrrr9rs1rngotrmorenewnrs1donerrrrs `` ` @@r/rz Expr.series s@T 9$$D TQ !NOO A) a t(((CAt}}aV,999Cy*D6** s8q=CtO56 6 R[t}A||||!--c3X11csbnnBx((*SSuuSS}}uuCI~ ".. + )IIaa(//QCa/H9;;;vvaa((  1b46k*+221aStZ[2\AyCCC661afr$w./0 0 ==AMM 1Q[[5Lt,D1d#**4Q$T*RBy444wwtQ'', =##AD#AB #QVVAvvx!8AvaffQ8At+<(<==eAqk)AX L %a !//QXDt/T!wwy4<"#gq4xotd{.K&L"LC!%!3!3A4d!3!SB"$'')a-%)%7%7StRV%7%W #q#%'')a-"!,),/FD*:;;%1a.(BGGIZZ\ !Q$N ++-7FF :r1~))  &6!!=!=adQU!=!VW WS )IIaa(00a08vvaa(( )H+I ' Av  s=>5T 4T $&U U" AU U UU"U32U3czddlm}|j|jcxurAnn>|dd}|j ||j ||||j ||Sddlm}ddlm }m }  |||\} } dd l m } m } dd lm}|| vrt|j || |j ||||j || |}|j!r"||d||zz |t"j$fzS|S|d d| | | |\}}|| vr|dkr|S|jj ||| }|j&|j(Dcgc]}|j+c}} | |j |d|z j-|j |d|z }| |j(d|j(dz z }| d|z }|j/d|}|r|j | |S|j!}t1t3j4|j+fd }t"j6}d}|D]u}|j9\}}|j;|rI|j |||dz }|r|j!rn%|j!rd}|||zzz }q||z }w|r|r|j | |S||zj | |S#|$r|cYSwxYwcc}w#|$r|}YwxYw)a Asymptotic Series expansion of self. This is equivalent to ``self.series(x, oo, n)``. Parameters ========== self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of ``x**n``, up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the ``bound`` parameter to give limit on rewriting coefficients in its normalised form. Examples ======== >>> from sympy import sin, exp >>> from sympy.abc import x >>> e = sin(1/x + exp(-x)) - sin(1/x) >>> e.aseries(x) (1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), (x, oo)))*exp(-x) >>> e.aseries(x, n=3, hir=True) -exp(-2*x)*sin(1/x)/2 + exp(-x)*cos(1/x) + O(exp(-3*x), (x, oo)) >>> e = exp(exp(x)/(1 - 1/x)) >>> e.aseries(x) exp(exp(x)/(1 - 1/x)) >>> e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x**2)*exp(exp(x)/x)*exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x**2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. >>> (1/x).aseries(x, n=8) 1/x Returns ======= Expr Asymptotic series expansion of the expression. Notes ===== This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either ``O(x ** (-n))`` or ``O(w ** (-n))`` where ``w`` belongs to the most rapidly varying expression of ``self``. References ========== .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion See Also ======== Expr.aseries: See the docstring of this function for complete details of this wrapper. rrrTr[rr)mrvrewrite)r;rdrr)boundc>t|jdS)Nr)ras_coeff_exponent)r-rs r/r.zExpr.aseries.. sQEXEXYZE[\]E^A_r1r/F)rrrrrrrrsympy.series.gruntzrrrsr;rdrrrr rr-rDras_leading_termrrr%r&rSrr?)r8rrrhirrrrrromrr;rdrr9r-logwr`rrrhas_ordr'exposnewrs @r/rz Expr.aseries! sr " ==AMM 1t,D99Q%--dAucBGGaP P'4 4|HB D, 7 !SV$,,Q5#>CCAs1vNAvvx51a4!QZZ999H # %T2q!, d 2:z ""1au"5Aaff5ff56A !&&AaC.88;@@AaCHItyy|chhqk12Q6Dqu:D KK1a  66!SY' ' FFHs}}QYY[17_` FFA--a0KE4yy|}}Qq}9tyy{YY["GdQWn%QG66!SY' 'A||As4y)){ K 06  s*3 L L*<=L/L'&L'/L:9L:cddlm}ddlm}t |}|d}|j ||j ||j ||j |d||zz||z S)zGeneral method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the "previous_terms". rrr) factorialr)rr(sympy.functions.combinatorial.factorialsrrrr)r8rrprevious_termsrr_xs r/ taylor_termzExpr.taylor_term sg "F AJ 3ZyyB$$R+00Q7<>> from sympy import sin, log, Symbol >>> from sympy.abc import x, y >>> sin(x).nseries(x, 0, 6) x - x**3/6 + x**5/120 + O(x**6) >>> log(x+1).nseries(x, 0, 5) x - x**2/2 + x**3/3 - x**4/4 + O(x**5) Handling of the ``logx`` parameter --- in the following example the expansion fails since ``sin`` does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): >>> e = sin(log(x)) >>> e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... >>> logx = Symbol('logx') >>> e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the ``logx`` parameter is used: >>> e = x**y >>> e.nseries(x, 0, 2) x**y >>> e.nseries(x, 0, 2, logx=logx) exp(logx*y) rj)rr)rrr)r8rrrrrrs r/nseriesz Expr.nseries sZx $+++K 9cSj;;q"a4;8 8%%a14d%C Cr1cDttd|jz)aA Return terms of series for self up to O(x**n) at x=0 from the positive direction. This is a method that should be overridden in subclasses. Users should never call this method directly (use .nseries() instead), so you do not have to write docstrings for _eval_nseries(). z The _eval_nseries method should be added to %s to give terms up to O(x**n) at x=0 from the positive direction so it is available when nseries calls it.)rFrr-r8rrrrs r/rzExpr._eval_nseriesF s+"*.*-1II .6#7 r1c$ddlm}|||||S)z Compute limit x->xlim. r)re)rtre)r8rxlimrres r/rez Expr.limitV s .T1dC((r1c8ddlm}|ddddjddlm}m}|j |r ||}n|}|jdk(r|Sd d lm }dd l m }dd l m } |} ||d n|}| d } tj} | j rS|j#|d | z|j%j'j)} | dz} | j rS| | j+|||} | j-|S)zDeprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. r)SymPyDeprecationWarningcompute_leading_termriSUz1.12)feature useinsteadissuerx) Piecewisepiecewise_foldrrrcrr)rrr)sympy.utilities.exceptionsrr$sympy.functions.elementary.piecewiserrr?rrrrsrdrrr rCrrrwpowsimptrigsimprr) r8rrrrrrPrrdr_logxrincrs r/rzExpr.compute_leading_term\ s G*(%+  $&R 88I !$'DD <<>Q K!>, $ uV}$Ahuull$$Q!D&t$<CCEMMOXXZC AIDll =((4Q(C""1%%r1rcFt|dkDr|}|D]}|j|||}|S|s|St|d}|jst d|z||j vr|S|j |||}|ddlm}||ddStd |d |d ) a Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples ======== >>> from sympy.abc import x >>> (1 + x + x**2).as_leading_term(x) 1 >>> (1/x**2 + x + x**2).as_leading_term(x) x**(-2) rrrzexpecting a Symbol but got %srTr;)r7combinezas_leading_term(, )) rNrrrrr_eval_as_leading_termsympy.simplify.powsimprrF)r8rrsymbolsr*rrrs r/rzExpr.as_leading_term s$ wHK GAJ {{ c,e`` where x can be any symbolic expression. rrr)rrr@rNr;r rS)r8rrr9r*prrs r/rzExpr.as_coeff_exponent sd 3 D! ~~a 1 q6Q;Q4##%DAqAv!t !&&yr1c lddlm}ddlm}|j |||}|d}|j ||r|j |||}|j|\}} ||jvr ttd|d|d |d ||j |||}|| fS) z Returns the leading term a*x**b as a tuple (a, b). Examples ======== >>> from sympy.abc import x >>> (1+x+x**2).leadterm(x) (1, 0) >>> (1/x**2+x+x**2).leadterm(x) (1, -2) rrrrcrrz) cannot compute leadterm(rz<). The coefficient should have been free of z but got ) rrrsrdrr?rrrrr) r8rrrrrdrrr*rs r/rz Expr.leadterm s ">  D 9 &M 55Q=s1vq!A""1%1  Z=A1a)LMN N FF1c!f !t r1c&tj|fS)z1Efficiently extract the coefficient of a product.rFr8rs r/r>zExpr.as_coeff_Mul suud{r1c&tj|fS)z3Efficiently extract the coefficient of a summation.)r rSrs r/rgzExpr.as_coeff_Add svvt|r1c ,ddlm}|||||||||S)z Compute formal power power series of self. See the docstring of the :func:`fps` function in sympy.series.formal for more information. r)fps)sympy.series.formalr) r8rrrhyperr=rfullrs r/rzExpr.fps s  ,4BUE8TBBr1c ddlm}|||S)zCompute fourier sine/cosine series of self. See the docstring of the :func:`fourier_series` in sympy.series.fourier for more information. r)fourier_series)sympy.series.fourierr )r8limitsr s r/r zExpr.fourier_series s 8dF++r1cD|jddt|g|i|S)Nr+T) setdefault_derivative_dispatch)r8r assumptionss r/rz Expr.diff s'z40#DB7BkBBr1c X|jdi|\}}|tj|zzS)Nr4)rr r)r8r8rrs r/_eval_expand_complexzExpr._eval_expand_complexs/&T&&// daood***r1c :d}|rmt|ddr`|jsTg}|jD]2}tj||fi|\}}||z}|j |4|r|j |}t||rt||di|}||k7r|dfS||fS)a  Helper for ``expand()``. Recursively calls ``expr._eval_expand_hint()``. Returns ``(expr, hit)``, where expr is the (possibly) expanded ``expr`` and ``hit`` is ``True`` if ``expr`` was truly expanded and ``False`` otherwise. FrDr4T)rRrGrDr3 _expand_hintr(r-r) rPr1r7r8rsargsrQarghitnewexprs r/rzExpr._expand_hints GD&"-dllEyy"//TCUC Vv  S!!  tyy%( 4 )gdD)2E2G$&c{r1c z ddlm j|||||||}  fd} jddr1| |D cgc]} | jd||d c} \} }| |z S jddr#| |\} }| |jd||d z S jd dr#| |\} }| jd||d |z Sd }t j | D]+} |}|s d |z}tj| |fd |i \} }- | } jddrtj| dfd |i \} } jddrtj| dfd |i \} } jddrtj| dfd |i \} }| |k(rn|t|}|jr|dkrtd|zg}tj| D]3}|jd\}}||z}|s |j!||z5t|} | Scc} w)z Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. r)fraction) power_base power_expr,rd multinomialbasicc6|jddS)NexactF)r)rrr8s r/r.zExpr.expand..3shq%))GU*CDr1fracF)r7modulusrnumerc|dk(ry|S)z Make multinomial come before mulr,mulzr4)r1s r/_expand_hint_keyz%Expr.expand.._expand_hint_keyOsu}Kr1r/ _eval_expand_r7Tr_eval_expand_multinomialr,_eval_expand_mulrd_eval_expand_logz*modulus must be a positive integer, got %srr4)rrr:rexpandrkeysr3rrrrrr%r&r>r()r8r7r#rrr,rdrrr8rP _fractionrrrr'r1use_hintrwasrrrr'rrs ` @r/r,z Expr.expand"s 4 iS 5  :D 99VU #&t_.,AHHA$A5A,.DAqQ3J YYw &T?DAqXQXXB4BEBB B YYw &T?DAq188@w@%@B B$  5::<-=>DT{H&- --dDMtMuM c ? Cyy.++4J;?JCHJayy&++,B37B;@Bayy&++,B37B;@Bas{  g&G%%A @7JLLE d+"///> t LLt, ,;D O.sH8c&ddlm}||g|i|S)z-See the integrate function in sympy.integralsr) integrate)sympy.integrals.integralsr2)r8rDrHr2s r/r2zExpr.integrates7////r1c$ddlm}|||||S)z,See the nsimplify function in sympy.simplifyr)r&)r;r&)r8 constants tolerancer r&s r/r&zExpr.nsimplifys5y)T::r1c$ddlm}||||S)z+See the separate function in sympy.simplifyr)expand_power_base)r7force)rr8)r8r7r9r8s r/separatez Expr.separates/ D>>r1c(ddlm}|||||||S)z*See the collect function in sympy.simplifyrr)rr)r8rr-r+r!distribute_order_termrs r/rz Expr.collects2tT45:OPPr1c&ddlm}||g|i|S)z(See the together function in sympy.polysr)together)sympy.polys.rationaltoolsr>)r8rDrHr>s r/r>z Expr.togethers6.t.v..r1c "ddlm}|||fi|S)z%See the apart function in sympy.polysr)apart)sympy.polys.partfracrA)r8rrDrAs r/rAz Expr.aparts.T1%%%r1cddlm}||S)z*See the ratsimp function in sympy.simplifyr)ratsimp)sympy.simplify.ratsimprD)r8rDs r/rDz Expr.ratsimps2t}r1c  ddlm}||fi|S)z+See the trigsimp function in sympy.simplifyr)r)sympy.simplify.trigsimpr)r8rDrs r/rz Expr.trigsimps4%%%r1c  ddlm}||fi|S)z*See the radsimp function in sympy.simplifyr)radsimp)rrI)r8rHrIs r/rIz Expr.radsimps2t&v&&r1c&ddlm}||g|i|S)z*See the powsimp function in sympy.simplifyrr)rr)r8rDrHrs r/rz Expr.powsimps2t-d-f--r1cddlm}||S)z+See the combsimp function in sympy.simplifyr)combsimp)sympy.simplify.combsimprL)r8rLs r/rLz Expr.combsimps4~r1cddlm}||S)z,See the gammasimp function in sympy.simplifyr) gammasimp)sympy.simplify.gammasimprO)r8rOs r/rOzExpr.gammasimps6r1c&ddlm}||g|i|S)z2See the factor() function in sympy.polys.polytoolsr)r)rr)r8rrDrs r/rz Expr.factor0d*T*T**r1c&ddlm}||g|i|S)z&See the cancel function in sympy.polysr)rw)rrw)r8rrDrws r/rwz Expr.cancelrRr1cr|jrt|ddr t||Sddlm}|||g|i|S)zReturn the multiplicative inverse of ``self`` mod ``g`` where ``self`` (and ``g``) may be symbolic expressions). See Also ======== sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert rTr)invert)rrRrrrU)r8rrrDrUs r/rUz Expr.inverts= >>gad;tQ' '0dA----r1c|}|js td|js3t|j dds!tdt |z|t vr|S|jx}sm|j\}}|dur4|j|tj|j|zzS|jdr|j|S|s|tjS|St|xsd}|jrt!tt#||St%|}||z}|j't(D cgc]} | j*} } | rt-t/| nd} | t/d |} n t1|| }| | z} d } |j | | zt3d | z}|j4r6|j*d k(r'|jdr t)dSt7d t!|}|dkDrd nd }|||z j | z}|dkr t9d|dkDr||z }n|dk(r ||dzr|n| z }|| z }t|j:|}t=|t3d | }|jr||St)t?|| S|s ||kDr|d z }t)||Scc} w)a,Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples ======== >>> from sympy import pi, E, I, S, Number >>> pi.round() 3 >>> pi.round(2) 3.14 >>> (2*pi + E*I).round() 6 + 3*I The round method has a chopping effect: >>> (2*pi + I/10).round() 6 >>> (pi/10 + 2*I).round() 2*I >>> (pi/10 + E*I).round(2) 0.31 + 2.72*I Notes ===== The Python ``round`` function uses the SymPy ``round`` method so it will always return a SymPy number (not a Python float or int): >>> isinstance(round(S(123), -2), Number) True z Cannot round symbolic expressionrTrzExpected a number but got %s:FrNrr znot computing with precisionrz)not expecting int(x) to round away from 0g?) rrrGrrrrrrrr rrrSrrrr_magrCrrrrrr}rrrFrrrH)r8rrxrrr-rdigits_to_decimalallowrXprecsdpsshiftextraxfrrdif2iprs r/rz Expr.roundsF {{>? ?yyA53ilBDD (]H((((>>#DAqU{wwqzAOOAGGAJ$>>>xx{wwqz!Y166 -A - 16N <<5Q+, , G!A%"#''%.1.Q.1).k#e*%D ;b%.CsOE#"S(^SSu c"en , <rgrr rrrr,r2r&r:rr>rArDrrIrrLrOrrwrUr __round__rh __classcell__rs@r/r3r3+sp,"$I#I ((T !2 !2F,L 8&'8:& '9 &'89% &9 &'8:&!'9!&'89%!&9!&'8:& '9 &'89% &9 &'8:& '9 & &'89% &9 &'8>*$+9$&'8=)%*9%&'8:& '9 &'89% &9 &'8?+#,9#&'8>*#+9# &'8=)5*95&'8<(5)954 >- &'8(9(&'8%9%&'8.9.&'8+9+! / C C1717fL\jXKZ%NGHOb   ".$ "8 D41l4l,A\'@DXt(BH~+@(:&4l!F"H47r (F,P]~CC<[:[[r1cyrr4rs r/rz"AtomicExpr._eval_is_algebraic_exprrr1c|Sr$r4rs r/rzAtomicExpr._eval_nseriesr_r1c$tddd|hS)Nrtrurvrwrr7s r/rzzAtomicExpr.expr_free_symbolss!!# &+'E  G v r1)r)rlrmrnrorrGr5rgrrrrrrrrrzrwrxs@r/rzrzvsQ IGI :$\r1rzc ddlm}m}m}t |j }|st jS t|||}|d|zz dk\rd|d|zz cxkrdksJJ|dz }|S#ttf$r:t|tt|jd|dz }YpwxYw)zReturn integer $i$ such that $0.1 \le x/10^i < 1$ Examples ======== >>> from sympy.core.expr import _mag >>> from sympy import Float >>> _mag(Float(.1)) 0 >>> _mag(Float(.01)) -1 >>> _mag(Float(1234)) 4 r)log10ceilrd5rWr)mathrrrdrrr rSrr OverflowErrorrrr)rrrrdr mag_first_digs r/rXrXs&% qssu:D vv JDt-.  R 1$T"m++1r11111   &JDwtzz2'>!?B!GHI JsA66AB?>B?ceZdZdZdZdZy)UnevaluatedExprz Expression that is not evaluated unless released. Examples ======== >>> from sympy import UnevaluatedExpr >>> from sympy.abc import x >>> x*(1/x) 1 >>> x*UnevaluatedExpr(1/x) x*1/x c Jt|}tj||fi|}|Sr$)r r3__new__)rrQrHrs r/rzUnevaluatedExpr.__new__s%smll3.v. r1c |jddr|jdjdi|S|jdS)Nr7Trr4)rrDr)r8r8s r/rzUnevaluatedExpr.doits< 99VT "$499Q<$$-u- -99Q< r1N)rlrmrnrorrr4r1r/rrs   r1rcL||}|j|k(xr|j|k(S)aReturn True if `func` applied to the `args` is unchanged. Can be used instead of `assert foo == foo`. Examples ======== >>> from sympy import Piecewise, cos, pi >>> from sympy.core.expr import unchanged >>> from sympy.abc import x >>> unchanged(cos, 1) # instead of assert cos(1) == cos(1) True >>> unchanged(cos, pi) False Comparison of args uses the builtin capabilities of the object's arguments to test for equality so args can be defined loosely. Here, the ExprCondPair arguments of Piecewise compare as equal to the tuples that can be used to create the Piecewise: >>> unchanged(Piecewise, (x, x > 1), (0, True)) True )r-rD)r-rDrXs r/ unchangedrs(2 d A 66T> ,affn,r1cLeZdZd dZedZdZd dZd dZdZ dZ d Z y) ExprBuilderNct|dstdj|||_|g|_n||_||_||r|j yyy)N__call__zop {} needs to be callable)rrroprD validatorvalidate)r8rrDrchecks r/__init__zExprBuilder.__init__s]r:&8??CD D <DIDI"  !u MMO(- !r1cl|Dcgc]$}t|tr|jn|&c}Scc}wr$)rmrbuild)rDr-s r/ _build_argszExprBuilder._build_argss.HLM1Z;7 Q>MMMs)1cr|jy|j|j}|j|yr$)rrrD)r8rDs r/rzExprBuilder.validates1 >> !  *r1c|j|j}|jr|r|j||j|Sr$)rrDrr)r8rrDs r/rzExprBuilder.builds= * >>e DNND !tww~r1c|jj||jr|r|j|jyyyr$)rDr(rr)r8rQrs r/append_argumentzExprBuilder.append_argument$s6  >>e DMM499 %$>r1cH|dk(r |jS|j|dz S)Nrr)rrD)r8items r/ __getitem__zExprBuilder.__getitem__)s% 1977N99T!V$ $r1c4t|jSr$)rHrr7s r/__repr__zExprBuilder.__repr__/s4::<  r1ct|jD]M\}}t|tr|j |}|*|f|zcSt |t |k(sJ|fcSyr$)rrDrmr search_indexid)r8elemr-rQrets r/search_elementzExprBuilder.search_element2s` *FAs#{+&&t,?4#:%CBtH$t +r1)NNTr ) rlrmrnrrtrrrrrrrr4r1r/rrs; NN  & % !r1rrZrVr|)rBrr)r>)rrrrr)K __future__rtypingrcollections.abcr functoolsrrrr rr r singletonr rr rr decoratorsrrrcacherintfuncrsortingrrrrrsympy.utilities.miscrrrsympy.utilities.iterablesrr mpmath.libmprrmpmath.libmp.libintmathr rr! collectionsr"r0r3rzrXrrrr,r[rr%rr}rrBrrrrr>rrrrrr4r1r/rs" $ &<<RR %@>>7-/# G=Q5*G=QG=QTz6t6r< d :-:33l4#CCr1