\hSSKJr SSKJrJrJrJrJr SSKJ r J r SSK J r SSK JrJr SSKJrJrJr SSKJrJrJr SSKJrJrJr SS KJr SS KJrJ r SS K!J"r"J#r#J$r$ SS K%J&r&J'r'J(r( SS K)J*r* SSK+J,r, SSK-J.r. SSK/J0r0J1r1 S#Sjr2Sr3S$Sjr4Sr5SS.Sjr6S%Sjr7Sr8S&Sjr9S&Sjr:S&Sjr;Sr\=r?\>r@\TUSTU5n[AUURB5$TVs/sH n[EUSS9PM snmSnU(a:UR75nUb'URF"T6(aSnOURI5n[R"U5Vs/sH n[EUSS9PM nn[K[,5[LRNnnUGHEnURQSS9unn[-[/U55U-n U Vs/sH nT/"U5PM n!nSn"THn#[U#[R5(a"U"(a[-[UU!55n!U"(+n"U"U!U#5n$U$cMHU#RV(d [YS5eU$un!n%n&n'U'(dI/n(U%H8n)U)ScU)Sn*O U)SU)S-n*U(R[[AU)S U*55 M: []U(6n+OU"U%5n+[EU"U!5SS9n![EU+SS9n+UU+R[U!5 GM> UU- nGMH UR5V Vs0sHupU [U6_M nn nU[LRNLaUUR^'Ub"UR5Hun,n-U-U-UU,'M Ub,UR5V,V-s0sHun,n-U,U"U-5_M nn,n-U(a/[UR5V,V-s/sH un,n-U,U--PM sn-n,6$U$s snfs snfs sn fs snn fs snn fs snfs snfs snfs snfs snfs snn fs sn-n,fs sn-n,f)a Collect additive terms of an expression. Explanation =========== This function collects additive terms of an expression with respect to a list of expression up to powers with rational exponents. By the term symbol here are meant arbitrary expressions, which can contain powers, products, sums etc. In other words symbol is a pattern which will be searched for in the expression's terms. The input expression is not expanded by :func:`collect`, so user is expected to provide an expression in an appropriate form. This makes :func:`collect` more predictable as there is no magic happening behind the scenes. However, it is important to note, that powers of products are converted to products of powers using the :func:`~.expand_power_base` function. There are two possible types of output. First, if ``evaluate`` flag is set, this function will return an expression with collected terms or else it will return a dictionary with expressions up to rational powers as keys and collected coefficients as values. Examples ======== >>> from sympy import S, collect, expand, factor, Wild >>> from sympy.abc import a, b, c, x, y This function can collect symbolic coefficients in polynomials or rational expressions. It will manage to find all integer or rational powers of collection variable:: >>> collect(a*x**2 + b*x**2 + a*x - b*x + c, x) c + x**2*(a + b) + x*(a - b) The same result can be achieved in dictionary form:: >>> d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False) >>> d[x**2] a + b >>> d[x] a - b >>> d[S.One] c You can also work with multivariate polynomials. However, remember that this function is greedy so it will care only about a single symbol at time, in specification order:: >>> collect(x**2 + y*x**2 + x*y + y + a*y, [x, y]) x**2*(y + 1) + x*y + y*(a + 1) Also more complicated expressions can be used as patterns:: >>> from sympy import sin, log >>> collect(a*sin(2*x) + b*sin(2*x), sin(2*x)) (a + b)*sin(2*x) >>> collect(a*x*log(x) + b*(x*log(x)), x*log(x)) x*(a + b)*log(x) You can use wildcards in the pattern:: >>> w = Wild('w1') >>> collect(a*x**y - b*x**y, w**y) x**y*(a - b) It is also possible to work with symbolic powers, although it has more complicated behavior, because in this case power's base and symbolic part of the exponent are treated as a single symbol:: >>> collect(a*x**c + b*x**c, x) a*x**c + b*x**c >>> collect(a*x**c + b*x**c, x**c) x**c*(a + b) However if you incorporate rationals to the exponents, then you will get well known behavior:: >>> collect(a*x**(2*c) + b*x**(2*c), x**c) x**(2*c)*(a + b) Note also that all previously stated facts about :func:`collect` function apply to the exponential function, so you can get:: >>> from sympy import exp >>> collect(a*exp(2*x) + b*exp(2*x), exp(x)) (a + b)*exp(2*x) If you are interested only in collecting specific powers of some symbols then set ``exact`` flag to True:: >>> collect(a*x**7 + b*x**7, x, exact=True) a*x**7 + b*x**7 >>> collect(a*x**7 + b*x**7, x**7, exact=True) x**7*(a + b) If you want to collect on any object containing symbols, set ``exact`` to None: >>> collect(x*exp(x) + sin(x)*y + sin(x)*2 + 3*x, x, exact=None) x*exp(x) + 3*x + (y + 2)*sin(x) >>> collect(a*x*y + x*y + b*x + x, [x, y], exact=None) x*y*(a + 1) + x*(b + 1) You can also apply this function to differential equations, where derivatives of arbitrary order can be collected. Note that if you collect with respect to a function or a derivative of a function, all derivatives of that function will also be collected. Use ``exact=True`` to prevent this from happening:: >>> from sympy import Derivative as D, collect, Function >>> f = Function('f') (x) >>> collect(a*D(f,x) + b*D(f,x), D(f,x)) (a + b)*Derivative(f(x), x) >>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), f) (a + b)*Derivative(f(x), (x, 2)) >>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), D(f,x), exact=True) a*Derivative(f(x), (x, 2)) + b*Derivative(f(x), (x, 2)) >>> collect(a*D(f,x) + b*D(f,x) + a*f + b*f, f) (a + b)*f(x) + (a + b)*Derivative(f(x), x) Or you can even match both derivative order and exponent at the same time:: >>> collect(a*D(D(f,x),x)**2 + b*D(D(f,x),x)**2, D(f,x)) (a + b)*Derivative(f(x), (x, 2))**2 Finally, you can apply a function to each of the collected coefficients. For example you can factorize symbolic coefficients of polynomial:: >>> f = expand((x + a + 1)**3) >>> collect(f, x, factor) x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + (a + 1)**3 .. note:: Arguments are expected to be in expanded form, so you might have to call :func:`~.expand` prior to calling this function. See Also ======== collect_const, collect_sqrt, rcollect cUR=(d2 U*R=(d [UR[55$N) is_Symbolboolatomsrxs N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/simplify/radsimp.pycollect..s1Q[[aRNNd  7T)binary)funcevaluateexactdistribute_order_termNc3f># UH&oR=(a URT;v M( g7fr')is_Powbase).0isymss r- collect..s#@%QXX0!&&D.0%s.1Fc:/nUHup#pEUb!Uupg[U5Hn[X&5nM UcBU[RLaUR U5 MSUR [ X#55 MoUR [ X#U-55 M [ U6$r')rangerrOneappendrr) termsproducttermratsymderivvarorder_s r-make_expression collect..make_expressions%* !Ds " uA%d0D&{!%%<NN4(NN3t>2s4S12&+G}r0c^URURSSp2nURSSHnXB:XaUS- nM[S5e [U[5(aURSm[ U4SjUR55(a [S5eTU:Xa$URU[ UR5-p1OO[U[5(aMX[U544$)Nrr6z(Improve MV Derivative support in collectc3,># UH oT:gv M g7fr')r:ss0s r-r=4collect..parse_derivative..s3Nq7Ns)expr variablesNotImplementedError isinstanceranylenr)rHrTrGrJrQrRs @r-parse_derivative!collect..parse_derivatives!::uq'915$Ax )>@@ %z**"B3DNN333)>@@Sy"iiT^^1D)Dez**8E?+++r0c>[RSp!USpCUR(Ga"[UR[ 5(aT"UR5up4O URnUR[R :Xa_URnUR(a[R UpGO&UR(aURSS9upg[U5UpOURR(a URnOURR5upgUR(aXgp!OURnO[U[5(a^URnUR(a[R UpONUR(aURSS9upg[U5UpO[U[ 5(a T"U5up4X1X$4$)aParses expression expr and outputs tuple (sexpr, rat_expo, sym_expo, deriv) where: - sexpr is the base expression - rat_expo is the rational exponent that sexpr is raised to - sym_expo is the symbolic exponent that sexpr is raised to - deriv contains the derivatives of the expression For example, the output of x would be (x, 1, None, None) the output of 2**x would be (2, 1, x, None). NT)rational) rrAr8rWr9rExp1r is_Rationalis_Mul as_coeff_Mul is_Number) rTrat_exposym_exposexprrHargcoefftailrZs r- parse_termcollect..parse_termsRUUD(Tu ;;;$))Z00/ : u yyAFF"hh??&'ffc8ZZ"%"2"2D"2"AKE&)$i8##88"hh335 ??).h#xxH c " "((C"#&&#x!...= "%d)Ux j ) )+D1LE//r0c>[R"U5n[U5[U5:agUVs/sH nT"U5PM nnUSSn/SSpTnUHup&pxUR(a US:XaUcM"[ [U55Hn X cM X uppU bSnU R U5cM*X:XdU cM4UcM9U R U5cMMTSLaX- nUcUnOXN:waSnO Xk:wdX:waMpUR X 5 SX ' M g UVs/sH o(dM UPM snX4U4$s snfs snf)zParse terms searching for a pattern. Terms is a list of tuples as returned by parse_terms; Pattern is an expression treated as a product of factors. NFr6T)r make_argsrYrbr@matchrB)rCpatternelemelems common_expo has_derive_rate_syme_ordjrEt_ratt_symt_ordexpo_fr4ris r-parse_expression!collect..parse_expression6sc --( u:G $4;>eqjU]s5z*Ax' 05-D ($(  4(4"^u/@!-!KK.: E>$)=D*2.2 $/#623K %~ ( UX.#'S+Z g.5j"'-2"B-u9L Lu=t.sD9 D>-D>r)deep)split_1z%Can not collect noncommutative symbolrP)0rr#r$dictziprr getcollectsubsitemsrWxreplacesetr rlhas_freer`add _new_rawargs as_coeff_mulalllistrrr3is_AddgetOr2argsr8r9rrrhasremoveOrrZeroargs_cncrreversedis_commutativeAttributeErrorrBrrA)0rTr<r2r3r4r5r;condrKnonsymsrepsrQrvkvurep_symsgsimplerLr|oarEb order_termsumma collecteddislikedrDcncrrC small_firstsymbolresultrprqrrmargsroeindexkeyvalrZris0 ` ` @@r-rrsl 4=D)1$dV!C E!C1GAJ!C ED DdD.JAC7!K7a%"9+a."97!KLM(,-1- TYYt_d"79"&.."d##;;t$ $!# , *HHQN++D11::d3CC *, , }t$A::t$T 88  ! NNANND$9!$<=D=IIaL%@%@@d75>**4"79 9$--&,400dFMP ;;  qA99!YY2&!q&OGAtT48MN&23567D[[99 II'%DdD$7LM%'( ([[ 4tU4IKAq$((# #6: ;d ae ,d ;DJYY[  !~~t$! ||~7:}}T7J K7J! qu -7JE K%d+QVVxI   /2GAJ"$(,-1A- F&*--+Xe_-"-o %eV4F!,,()PQQ7=4uk9!E %7? $QA $QQA Sa!_5 !&  KE+E2E)/%*@uM)%e<% ''.9>  HKN)2(9:(9CG(9I:qvv# !%%!)HC :-IcN* +4??+<>+Y__->?->cSW->?@@O F"L-/,H2' < L .H;>@sSY %YYY(>> from sympy.simplify import rcollect >>> from sympy.abc import x, y >>> expr = (x**2*y + x*y + x + y)/(x + y) >>> rcollect(expr, y) (x + y*(x**2 + x + 1))/(x + y) See Also ======== collect, collect_const, collect_sqrt )is_Atomr __class__rrcollectrr)rTvarsrfs r-rrsb( ||488T? ~~ J  4t 4 JK ;;4& &K KsA:cUc[RnUR5up [5n[R "U5HnUR 5SHnUR(dMUR(a5URR(aURRS:XdU[RLdMqURU5 M M [U/UQ7SS06nX:gnU(dSn[![#[R "U555n [%U 5HupUR 5upU Hcn U R(a5U RR(aU RRS:XdU [RLdM^US- n O X==U-ss'M U(dU(d [ U 6/n ['U 5U4$X&-$)aReturn expr with terms having common square roots collected together. If ``evaluate`` is False a count indicating the number of sqrt-containing terms will be returned and, if non-zero, the terms of the Add will be returned, else the expression itself will be returned as a single term. If ``evaluate`` is True, the expression with any collected terms will be returned. Note: since I = sqrt(-1), it is collected, too. Examples ======== >>> from sympy import sqrt >>> from sympy.simplify.radsimp import collect_sqrt >>> from sympy.abc import a, b >>> r2, r3, r5 = [sqrt(i) for i in [2, 3, 5]] >>> collect_sqrt(a*r2 + b*r2) sqrt(2)*(a + b) >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r3) sqrt(2)*(a + b) + sqrt(3)*(a + b) >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5) sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b) If evaluate is False then the arguments will be sorted and returned as a list and a count of the number of sqrt-containing terms will be returned: >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5, evaluate=False) ((sqrt(3)*a, sqrt(5)*b, sqrt(2)*(a + b)), 3) >>> collect_sqrt(a*sqrt(2) + b, evaluate=False) ((b, sqrt(2)*a), 1) >>> collect_sqrt(a + b, evaluate=False) ((a + b,), 0) See Also ======== collect, collect_const, rcollect rrNumbersFr6)rr3as_content_primitiverr rlr is_numberr8rr_qr ImaginaryUnitr collect_constrr enumeratetuple)rTr3rgrrmdhitnradrr;rrcis r- collect_sqrtrsjR$--++-KE 5D ]]4 aA{{{HH!2!2quuww!|( !! d1T151A )C GCMM!,-.dODAJJLEA99!3!3A aoo-AID  Gu G$tJ>> from sympy.simplify.radsimp import collect_abs >>> from sympy.abc import x >>> collect_abs(abs(x + 1)/abs(x**2 - 1)) Abs((x + 1)/(x**2 - 1)) >>> collect_abs(abs(1/x)) Abs(1/x) cUR5up/n/nUHn[U[5(a URURS5 M8[U[ 5(aq[UR [5(aRURR(a7URUR RSUR-5 MURU5 M [U5S:a[SU55(dU$[U6n[U5nU/nURU5 UR[5(dURU5 [U6$[U[5(d [USS9nXxS'[U5 URU5 [R"X(+S9$)Nrrc3|# UH2n[U[5(dMURRv M4 g7fr')rWrr is_negative)r:r;s r-r=,collect_abs.._abs..^s'!UQ*QPSBT"3!%%"3"3Qs<<Fr3)r)rrWr rBrrr9ris_realrYrXrextendrr _from_args) mulrrrrr;absargArs r-_abscollect_abs.._absSsB   A!S!!#As## 1663(?(?AEEMMQ./  q6A:c!UQ!UUUJa Ks AuuSzz KKO: !S!!FU+AQ B~~d6::r0c"[U[5$r')rWrr+s r-r.collect_abs..ps *Q$r0c>T"U5$r'rPr,rs r-r.rqs $q'r0c"[U[5$r')rWrr+s r-r.rrs jC(r0c>T"U5$r'rPrs r-r.rss d1gr0)replace)rTrs @r- collect_absrEs1;8 <<$ "7 ( r0)rc ^^UR(dU$SnU(daSn[5nURHDn[R"U5H'nUR (dMUR U5 M) MF O [U5nU(d$UVs/sHofR(aMUPM nn[[U55nUGHn[[5n[U5n[R"U5Hn[U5n U RU5upU R(adU R R#5mU R m[%UU4SjT55(d#XvR'U R)55 MU[*R,R'U5 M /n Sn Sn[U5HnXnU[*R,LaU R/U5 M-[1U5S:a([U6nSn U(aX`:waUR'U5 OUSnU(a?UR(a.UR(aU R'[3XSS95 SnMU R'X-5 M U (dGMU(a [5U 6nO[U 6nUR(aGM U$ U$s snf)aA non-greedy collection of terms with similar number coefficients in an Add expr. If ``vars`` is given then only those constants will be targeted. Although any Number can also be targeted, if this is not desired set ``Numbers=False`` and no Float or Rational will be collected. Parameters ========== expr : SymPy expression This parameter defines the expression the expression from which terms with similar coefficients are to be collected. A non-Add expression is returned as it is. vars : variable length collection of Numbers, optional Specifies the constants to target for collection. Can be multiple in number. Numbers : bool Specifies to target all instance of :class:`sympy.core.numbers.Number` class. If ``Numbers=False``, then no Float or Rational will be collected. Returns ======= expr : Expr Returns an expression with similar coefficient terms collected. Examples ======== >>> from sympy import sqrt >>> from sympy.abc import s, x, y, z >>> from sympy.simplify.radsimp import collect_const >>> collect_const(sqrt(3) + sqrt(3)*(1 + sqrt(2))) sqrt(3)*(sqrt(2) + 2) >>> collect_const(sqrt(3)*s + sqrt(7)*s + sqrt(3) + sqrt(7)) (sqrt(3) + sqrt(7))*(s + 1) >>> s = sqrt(2) + 2 >>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7)) (sqrt(2) + 3)*(sqrt(3) + sqrt(7)) >>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7), sqrt(3)) sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2) The collection is sign-sensitive, giving higher precedence to the unsigned values: >>> collect_const(x - y - z) x - (y + z) >>> collect_const(-y - z) -(y + z) >>> collect_const(2*x - 2*y - 2*z, 2) 2*(x - y - z) >>> collect_const(2*x - 2*y - 2*z, -2) 2*x - 2*(y + z) See Also ======== collect, collect_sqrt, rcollect FTc3># UH;nUT;=(a* TUR=(a TUR(+v M= g7fr') is_Integer)r:rfnowfwass r-r= collect_const..sJ:48q9+a););+Q**A++48sAAr6r)sign)rrrrrlrrrrbrrrr r divis_onefactorscopyrXrBas_exprrrArrYrr )rTrrrecurserrrrCFvfrrrrunevalrrrs @@r-rrvs(| ;; G uA]]1%;;;HHQK& t} 34a{{43  D D! QZt$A A559DAxx yy~~'yy:48:::HOOAIIK0 !%%L   "% AAAEEz A1vzGqyKKNaD 1;;188 K489 AC + . 3'.Dz;;; Kkj Kq4s K.Kc ^^^^^^^SSKJm SSKJn [ S5mU4SjmS U4SjjmUUUUUU4Sjm[ UT5(d3UR "URVs/sHn[UTTS 9PM sn6$UR5upPUR5n[U5n[T"U55upxXgU4:wGaoUR(dxXx4n U"USS 9nU"USS 9n[[US U- 55n [U RR!5V V s/sH upX-PM sn n 6n [U 5upxXgU4:XaU upx[#U5nUR$(dUR&(a[[)[US U- 555upUR$(d"UR+5UR+5::acX4Vs/sH o"U5PM snupxUR,(a7URSR$(aUR "UR6nU[US U- 5-$s snfs sn n fs snf) a Rationalize the denominator by removing square roots. Explanation =========== The expression returned from radsimp must be used with caution since if the denominator contains symbols, it will be possible to make substitutions that violate the assumptions of the simplification process: that for a denominator matching a + b*sqrt(c), a != +/-b*sqrt(c). (If there are no symbols, this assumptions is made valid by collecting terms of sqrt(c) so the match variable ``a`` does not contain ``sqrt(c)``.) If you do not want the simplification to occur for symbolic denominators, set ``symbolic`` to False. If there are more than ``max_terms`` radical terms then the expression is returned unchanged. Examples ======== >>> from sympy import radsimp, sqrt, Symbol, pprint >>> from sympy import factor_terms, fraction, signsimp >>> from sympy.simplify.radsimp import collect_sqrt >>> from sympy.abc import a, b, c >>> radsimp(1/(2 + sqrt(2))) (2 - sqrt(2))/2 >>> x,y = map(Symbol, 'xy') >>> e = ((2 + 2*sqrt(2))*x + (2 + sqrt(8))*y)/(2 + sqrt(2)) >>> radsimp(e) sqrt(2)*(x + y) No simplification beyond removal of the gcd is done. One might want to polish the result a little, however, by collecting square root terms: >>> r2 = sqrt(2) >>> r5 = sqrt(5) >>> ans = radsimp(1/(y*r2 + x*r2 + a*r5 + b*r5)); pprint(ans) ___ ___ ___ ___ \/ 5 *a + \/ 5 *b - \/ 2 *x - \/ 2 *y ------------------------------------------ 2 2 2 2 5*a + 10*a*b + 5*b - 2*x - 4*x*y - 2*y >>> n, d = fraction(ans) >>> pprint(factor_terms(signsimp(collect_sqrt(n))/d, radical=True)) ___ ___ \/ 5 *(a + b) - \/ 2 *(x + y) ------------------------------------------ 2 2 2 2 5*a + 10*a*b + 5*b - 2*x - 4*x*y - 2*y If radicals in the denominator cannot be removed or there is no denominator, the original expression will be returned. >>> radsimp(sqrt(2)*x + sqrt(2)) sqrt(2)*x + sqrt(2) Results with symbols will not always be valid for all substitutions: >>> eq = 1/(a + b*sqrt(c)) >>> eq.subs(a, b*sqrt(c)) 1/(2*b*sqrt(c)) >>> radsimp(eq).subs(a, b*sqrt(c)) nan If ``symbolic=False``, symbolic denominators will not be transformed (but numeric denominators will still be processed): >>> radsimp(eq, symbolic=False) 1/(a + b*sqrt(c)) r)Expr)signsimpza:d A:Dc>T upp4pVpx[U5S:Xag[[[XQXb/UV V s/sH oHoPM M sn n 555n [ U5U-[ U5U-- R U 5$[U5S:Xa[[[XQXbXs/UV V s/sH oHoPM M sn n 555n [ U5U-[ U5U--[ U5U-- S[ U5-[ U5-U-U-XQS--- XbS--- XsS----R U 5$[U5S:XGa[[[XQXbXsX/UV V s/sH oHoPM M sn n 555n [ U5U-[ U5U--[ U5U-- [ U5U-- S[ U5-[ U5-U-U-XQS--- XbS--- S[ U5-[ U5-U-U-- XsS---XS----S[ U5-[ U5-[ U5-[ U5-U-U-U-U-US-US---SU-U-US--US--- SU-U-US--US--- SU-U-US--US--- US-US---SU-U-US--US--- SU-U-US--US--- US-US---SU-U-US--US--- US-US----R U 5$[U5S:Xa[ USS5$[ es sn n fs sn n fs sn n f)Nrir6r)rYrrrrrrV) rtermsrrrrrBCDr;rvrr<s r-_numradsimp.._numNs"&aA! v;! S!6/M6a1a16/MNOPD GAIQ !88D> 2 v;! S!a!35SAQRAaQRa5STUVD !WQYa "T!WQY .47471B11DQ1Fa41O dF2!tV2 &htn -[A S!aA!9v;Yv!WXRSAWXAv;YZ[\D!WQYa*T!WQY6aBQtAwYtTUwEVWXEXYZEZqD&FT6F"$%d1gId1g$5a$7$9F:<=dFFCQ$FDGDG+DG3DG;A=a?A!Cad1a4iO!Aad 1a4 "#A#a%1*QT/245aCE!Q$Jq!tODFGd1a4iP!Aad 1a4 "#A#a%1*QT/245qDAI>@A!Aad 1a4P1QT  %HTN  + [A q ! % %% %'0N6T UR(dgURnUR(aURS:XdT(a[ U5S:XagU(agSnUR(a URnO%T(a[ U5nUR (aUnUS:wa[ US5R (agg)NFrTr6)r8rr_rdenomrr)rlog2rrsymbolics r-ispow2radsimp..ispow2hsxx EE ==QSSAXeAh!m A}}CC!H<<AAv#a)..r0c ^ >SSKJn UR(aU$[UT5(d0UR"UR Vs/sH nT"U5PM sn6$[ U5up4UR(aUR(aU$UR(dEUR"UR Vs/sH nT"U5PM sn6n[UT"SU- 55$U[RLa[UT"SU- 55$UR(a,[UR Vs/sHnT"SU- 5PM sn6$T(dUR(aU$T"U5(aE[[UR55[UR 5-nXT:wa T"SU- 5$OiUR"(aXUR R$(dURR&(a"T"SUR- 5UR -$UR((d@T"U5(d3SUR"UR Vs/sH nT"U5PM sn6- $Sn[+U5nUR(aSU- $UR,(a4U"U5nUR.(aUR1U5(aSU- $[3[45n[6R8"U5Hn /n /n [:R8"U 5Hn T"U SS9(aVU R=U R [R>La U ROU RSU R --5 MeU [R@La!U R=[RB5 MU R=U 5 M U[E[GU 55R=[;U 65 M [5[G[5URI5555n U V Vs/sHup[;U 6[7U64PM n n n[KU 5U SS[RLaSOS- nUS:aGOUT:aSnO[KU 5S:as[MS U 55(aY[O[R[6RP"U VVs/sHunn[U5U-PM snn55unnUU-nOSnOuSS K)J*nJ+n U"T"U 55nUU-nUU-nU"[+U5TS 9nURY[RZ[\[^5(aU$UR(aOGMU(dU$[USU- 5$s snfs snfs snfs snfs snn fs snnf) Nr) nsimplifyr6T)rrFrc3j# UH)upUR=(a US-Rv M+ g7f)rN)rr_)r:r,ys r-r=*radsimp..handle..s(Nvtqq||:A(:(::vs13)powsimp powdenest)force)0sympy.simplify.simplifyrrrWr2rfractionrrrAr` free_symbolsr"rr9numerrr8 is_integer is_positiverrrrbequalsrrr rlrrBHalfr NegativeOnerrrrYrrad_rationalizersympy.simplify.powsimprrrrrr)rTrrnrd2keep_drrp2otherr;rrvrr,rndrrnumrrhandler max_termsrs r-rradsimp..handlezsS 6 <<KD$''99$))<)Qvay)<= =~ 99KAFF3FqF34A#Avac{3 3 aee^#Avac{3 3 XX#166%B6afQqSk6%BC CANNK !99DL)5<7Bwad|# XX155++qvv/A/A!AFF(#QUU* *F1IIQVV8AfQi899 9 QK 99Q3J ;;1B|| ! t #D)I]]1%q)Aad+ AEEQVVO!&&!AEE'ARSaoo- !--0 Q *% ,-44S%[A&'$y'8"9:;F5;>/56vtq!av6489EBGA!D A$v,'C HA HA(1+X6AuuQVVS#&& yyadK1Q3''G=4 &C9F=$7s$V8V/VV 'V#V))rrrr6F)sympy.core.exprrr rrrWr2rradsimp as_coeff_Addnormalr rr rrrrrbrr count_opsr`)rTrrrrrgoldrrwasurrn2rr;rrrrr<s `` @@@@@r-r!r!sX%0 9 D&4$l(l(\ dD ! !yyW[W`W`aW`RS71x9MW`abb##%KE ;;=D 4.C F4L !DA !f}yy&CU+AU+A(AaC01A AIIOO4E"F4EDA144E"FGAA;DA!f} qM ;;!((i(8AaC(@ABFB|| !++- ?.0X6X X688q 3 3A #Aqs+ ++1b#G7s#I I IcUR(dX4$[U5up#nU[U5-n[X4- U-5n[US-US-- 5n[ X5$)a8 Rationalize ``num/den`` by removing square roots in the denominator; num and den are sum of terms whose squares are positive rationals. Examples ======== >>> from sympy import sqrt >>> from sympy.simplify.radsimp import rad_rationalize >>> rad_rationalize(sqrt(3), 1 + sqrt(2)/3) (-sqrt(3) + sqrt(6)/3, -7/9) r)r split_surdsrrr)rdenrrrs r-rrs^ ::x#GA! $q' A AE3; C 1a4!Q$; C 3 $$r0c[U5n//p2[R"U5GHnUR(GagUR(d[ U[ 5(Ga@UR5upVUR(aU[RLaURU5 MU(aEUR5(aUR[XV*55 MURU5 MUR[XV*55 MUR(aURU5 GMU(dNUR(a=UR!5upxUS:waURU5 URU5 GMiURU5 GM}UR"(aZUR$(dIUR&S:waURUR&5 URUR(5 GMURU5 GM [USU(+06[USU(+064$)aReturns a pair with expression's numerator and denominator. If the given expression is not a fraction then this function will return the tuple (expr, 1). This function will not make any attempt to simplify nested fractions or to do any term rewriting at all. If only one of the numerator/denominator pair is needed then use numer(expr) or denom(expr) functions respectively. >>> from sympy import fraction, Rational, Symbol >>> from sympy.abc import x, y >>> fraction(x/y) (x, y) >>> fraction(x) (x, 1) >>> fraction(1/y**2) (1, y**2) >>> fraction(x*y/2) (x*y, 2) >>> fraction(Rational(1, 2)) (1, 2) This function will also work fine with assumptions: >>> k = Symbol('k', negative=True) >>> fraction(x * y**k) (x, y**(-k)) If we know nothing about sign of some exponent and ``exact`` flag is unset, then the exponent's structure will be analyzed and pretty fraction will be returned: >>> from sympy import exp, Mul >>> fraction(2*x**(-y)) (2, x**y) >>> fraction(exp(-x)) (1, exp(x)) >>> fraction(exp(-x), exact=True) (exp(-x), 1) The ``exact`` flag will also keep any unevaluated Muls from being evaluated: >>> u = Mul(2, x + 1, evaluate=False) >>> fraction(u) (2*x + 2, 1) >>> fraction(u, exact=True) (2*(x + 1), 1) r6r3)rrrlrr8rWr as_base_exprrrrB is_constantrrr`as_numer_denomr_rpr) rTr4r rrErexrrs r-r r stp 4=Dr5 d#    DKK:dC3H3H$$&EA~~&LLO~~'' SC[1 T*LLQ- T"ryy**,6LLO Q T"   doovv{ TVV$ LL LL 7$8  *E *C,KU,K KKr0c[XS9S$)Nr4rr rTr4s r-r r u D &q ))r0c[XS9S$)Nr3r6r4r5s r-rryr6r0c *UR"SSS0UD6$)NfracTrP)expand)rThintss r-fraction_expandr<}s ;; *D *E **r0c f[XRSS5S9up#UR"SSS0UD6U- $)Nr4Fr3r TrPr rr:rTr;rrs r- numer_expandr@s5 D '5 9 :DA 88 ($ (% (1 ,,r0c d[XRSS5S9up#X#R"SSS0UD6- $)Nr4Fr3rTrPr>r?s r- denom_expandrBs3 D '5 9 :DA xx,d,e, ,,r0c.[UR[S9nUVs/sHo"R5PM nnUVs/sH o"SR(dMUSS-PM" nnUR [S9 [ U6upVnUnU(dK[U5S:a<UVs/sHo"U- PM n nU Vs/sH o"S:wdM UPM n n[ U 6upnXZ-n//pUHupUR(aeUR[R:XaGURnX;a!U RU [X- 5-5 MfU RX-5 M{U RX-5 M [U 6n[U 6nUUU4$s snfs snfs snfs snf)a Split an expression with terms whose squares are positive rationals into a sum of terms whose surds squared have gcd equal to g and a sum of terms with surds squared prime with g. Examples ======== >>> from sympy import sqrt >>> from sympy.simplify.radsimp import split_surds >>> split_surds(3*sqrt(3) + sqrt(5)/7 + sqrt(6) + sqrt(10) + sqrt(15)) (3, sqrt(2) + sqrt(5) + 3, sqrt(5)/7 + sqrt(10)) )rr6r)sortedrrrar8sort _split_gcdrYrrrr9rBrr )rTrr, coeff_mulssurdsrb1b2g2b1ng1a1va2vrrQs1rrs r-r*r*s[ $))!1 2D,01Dq.."DJ1( 8jaDKKWQqT1WjE 8 JJ#J$E"IA2 B #b'Q,BqsB(#Qaq#( #&  T2 88Bx 1T"%[=) 13 JJqsO S A S A q!8O/2 8  (s"FF FF 3 FFcUSnU/n/nUSSH:n[X5nUS:XaURU5 M'UnURU5 M< XU4$)a< Split the list of integers ``a`` into a list of integers, ``a1`` having ``g = gcd(a1)``, and a list ``a2`` whose elements are not divisible by ``g``. Returns ``g, a1, a2``. Examples ======== >>> from sympy.simplify.radsimp import _split_gcd >>> _split_gcd(55, 35, 22, 14, 77, 10) (5, [55, 35, 10], [22, 14, 77]) rr6N)r!rB)rrrIrJr,rMs r-rFrFs` !A B B qrU Y 7 IIaLA IIaL  "9r0)NNFTr')Trr)D collectionsr sympy.corerrrrrsympy.core.addr r sympy.core.assumptionsr sympy.core.exprtoolsr r sympy.core.functionrrrsympy.core.mulrrrsympy.core.numbersrrrsympy.core.parametersrsympy.core.sortingrrsympy.core.symbolrrrsympy.functionsrrr$sympy.functions.elementary.complexesr sympy.polysr!sympy.simplify.sqrtdenestr"sympy.utilities.iterablesr#r$rrrrrr!rr r rr<r@rB expand_numer expand_denomexpand_fractionr*rFrPr0r-res#770.3GGBB113822**404 ~B>L^.b(,EPC,L%,XLv**+- -   !&Rr0