ЦiSSKJr SSKJr SSKJr SSKJr SSK J r SSK J r J r JrJrJrJrJrJr SSKJrJrJr SSKJr SS KJrJrJrJr SS KJ r SS K!J"r" SS K#J$r$ SS K%J&r& SSK'J(r(J)r) SSK*J+r+ SSK,J-r- SSK.J/r/J0r0J1r1J2r2J3r3 SSK4J5r5 SSK6J7r7J8r8 SSK9J:r: "SS\ 5r;"SS\;5r<"SS\5r="SS\;\=S9r>Sr?"SS \ 5r@"S!S"\ 5rA\S#5rBg$)%)product)Tuple)Add)cacheit)Expr)FunctionArgumentIndexError expand_log expand_mul FunctionClass PoleErrorexpand_multinomialexpand_complex) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRationalpiI)global_parameters)Pow)Ge)S)WildDummy)sympify) factorial)arg unpolarifyimreAbs)sqrt) multiplicity perfect_power) factorintc\rSrSrSr\R 4r\S5r SSjr Sr \S5r Sr SrS rS rS rS rS rSrSrSrg)ExpBase$Tc.URR$N)expkindselfs e/var/www/html/ai-image-ml/venv/lib/python3.13/site-packages/sympy/functions/elementary/exponential.pyr/ ExpBase.kind)sxx}}c[$)z- Returns the inverse function of ``exp(x)``. logr1argindexs r2inverseExpBase.inverse-  r4c@UR(dU[R4$URnURnU(d"U*R(dUR 5nU(a"[RUR U*54$U[R4$)z Returns this with a positive exponent as a 2-tuple (a fraction). Examples ======== >>> from sympy import exp >>> from sympy.abc import x >>> exp(-x).as_numer_denom() (1, exp(x)) >>> exp(x).as_numer_denom() (exp(x), 1) )is_commutativerOner. is_negativecould_extract_minus_signfunc)r1r.neg_exps r2as_numer_denomExpBase.as_numer_denom3sr ""; hh//11224G 55$))SD/) )QUU{r4c URS$)z' Returns the exponent of the function. r)argsr0s r2r. ExpBase.expMs yy|r4cHURS5[UR64$)z' Returns the 2-tuple (base, exponent). )rBrrGr0s r2 as_base_expExpBase.as_base_expTsyy|S$))_,,r4cTURURR55$r-)rBr.adjointr0s r2 _eval_adjointExpBase._eval_adjointZsyy))+,,r4cTURURR55$r-)rBr. conjugater0s r2_eval_conjugateExpBase._eval_conjugate]yy++-..r4cTURURR55$r-)rBr. transposer0s r2_eval_transposeExpBase._eval_transpose`rUr4cURnUR(a$UR(agUR(agUR(aggNTF)r. is_infiniteis_extended_negativeis_extended_positive is_finiter1r s r2_eval_is_finiteExpBase._eval_is_finitecs9hh ??'''' == r4cUR"UR6nURUR:XaLURRnU(agURR(a[ U5(agggUR$r[)rBrGr.is_zero is_rationalr)r1szs r2_eval_is_rationalExpBase._eval_is_rationalmsc IItyy ! 66TYY  A""y||(4"== r4c:UR[RL$r-)r.rNegativeInfinityr0s r2 _eval_is_zeroExpBase._eval_is_zeroxsxx1----r4cdUR5up#[R"[X#SS9U5$)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)rKr _eval_power)r1otherbes r2rqExpBase._eval_power{s,!s1%8%@@r4c ^SSKJn SSKJn TRSnUR (a;UR (a*[R"U4SjUR55$[XC5(a=UR (a,U"TRUR5/URQ76$TRU5$)Nr)Product)Sumc3F># UHnTRU5v M g7fr-)rB).0xr1s r2 1ExpBase._eval_expand_power_exp..s?h ! hs!) sympy.concrete.productsrwsympy.concrete.summationsrxrGis_Addr>rfromiter isinstancerBfunctionlimits)r1hintsrwrxr s` r2_eval_expand_power_expExpBase._eval_expand_power_exps31iil ::#,,<<?chh?? ?  ! !c&8&8499S\\2@SZZ@ @yy~r4NrJ)__name__ __module__ __qualname____firstlineno__ unbranchedrComplexInfinity_singularitiespropertyr/r:rDr.rKrOrSrXrarhrlrqr__static_attributes__rr4r2r*r*$ssJ'')N  4 - -// !.A r4r*c>\rSrSrSrSrSrSrSrSr Sr S r S r g ) exp_polara Represent a *polar number* (see g-function Sphinx documentation). Explanation =========== ``exp_polar`` represents the function `Exp: \mathbb{C} \rightarrow \mathcal{S}`, sending the complex number `z = a + bi` to the polar number `r = exp(a), \theta = b`. It is one of the main functions to construct polar numbers. Examples ======== >>> from sympy import exp_polar, pi, I, exp The main difference is that polar numbers do not "wrap around" at `2 \pi`: >>> exp(2*pi*I) 1 >>> exp_polar(2*pi*I) exp_polar(2*I*pi) apart from that they behave mostly like classical complex numbers: >>> exp_polar(2)*exp_polar(3) exp_polar(5) See Also ======== sympy.simplify.powsimp.powsimp polar_lift periodic_argument principal_branch TFcD[[URS55$Nr)r.r#rGr0s r2 _eval_Absexp_polar._eval_Abss2diil#$$r4c0[URS5nU[*:*=(d U[:nU(aU$[ URS5R U5nUS:a[U5S:a [ U5$U$![a SnN`f=f)z-Careful! any evalf of polar numbers is flaky rT)r"rGr TypeErrorr. _eval_evalfr#)r1precibadress r2rexp_polar._eval_evalfs tyy|  8%q2vC K$))A,++D1 q5RWq[c7N  C sB BBcDURURSU-5$r)rBrG)r1rrs r2rqexp_polar._eval_powersyy1e+,,r4cBURSR(agg)NrT)rGis_extended_realr0s r2_eval_is_extended_real exp_polar._eval_is_extended_reals 99Q< ( ( )r4cvURSS:XaU[R4$[R U5$r)rGrr?r*rKr0s r2rKexp_polar.as_base_exps1 99Q<1 ; ""4((r4rN) rrrr__doc__is_polar is_comparablerrrqrrKrrr4r2rrs-#JHM% -)r4rc\rSrSrSrSrg)ExpMetac[URR;ag[U[5=(a UR [ RL$)NT)r. __class____mro__rrbaserExp1)clsinstances r2__instancecheck__ExpMeta.__instancecheck__s8 ($$,, ,(C(DX]]aff-DDr4rN)rrrrrrrr4r2rrsEr4rc\rSrSrSrSSjrSr\S5r\ S5r \ \ S55r SSjrS rS rS rS rS rSSjrSrSSjrSrSrSrSrSrSrg)r.z The exponential function, :math:`e^x`. Examples ======== >>> from sympy import exp, I, pi >>> from sympy.abc import x >>> exp(x) exp(x) >>> exp(x).diff(x) exp(x) >>> exp(I*pi) -1 Parameters ========== arg : Expr See Also ======== log c(US:XaU$[X5e)z0 Returns the first derivative of this function. rJ)r r8s r2fdiff exp.fdiffs q=K$T4 4r4cSSKJnJn URSnUR(Ga4[ [ R-nXEU*4;a[ R$UR"[[ -5nU(aU"URSU-55(aU"URU55(a[ R$U"URU55(a[ R$U"URU[ R -55(a[ *$U"URU[ R -55(a[ $gggg)Nr)askQ)sympy.assumptionsrrrGis_MulrrInfinityNaNas_coefficientrintegerevenr?odd NegativeOneHalf)r1 assumptionsrrr Ioocoeffs r2 _eval_refineexp._eval_refines,iil :::AJJ,CSDk!uu &&r!t,Eqyy5)**166%=)) uu QUU5\** }},QVVEAFFN344 !r QUU5166>233 4+ r4c SSKJn SSKJn SSKJn SSKJn [X5(aUR"5$[R(a[[RU5$UR(aU[R La[R $UR"(a[R$$U[R$La[R$U[R&La[R&$U[R(La[R*$GO@U[R,La[R $[U[.5(aUR0S$[X5(a/U"[UR25[UR455$[X5(aUR6"U5$UR8(GaUR:"[<[>-5nU(aSU-R@(aURB(a[R$$URD(a[RF$U[RH-RB(a[>*$U[RH-RD(a[>$O-5$URL"5uphU[R([R&4;aURN(aU[R(LaU*n[QU5R"(a#U[R*La[R $[QU5RR(a,[UU5[R*La[R,$[QU5RV(a[R*$gU/Sp[XRZ"U5HZn U"U 5n [U [.5(aU cU R0Sn M4 gU R\(aU R_U 5 MZ g U (a U [YU 6-$S$UR`(a/n /nSnUR0HnU[R$LaUR_U5 M)U"U5n[UU5(aHUR0SU:wa"UR_UR0S5 S nMwUR_U5 MU R_U5 M U (dU(a[YU 6U"[cU6SS 9-$UR"(a[R$$g) Nr AccumBounds) MatrixBaseSetExpr logcombinerrJFTro)2sympy.calculusrsympy.matrices.matrixbasersympy.sets.setexprrsympy.simplify.simplifyrrr.r exp_is_powrrr is_Numberrrdr?rrkZerorr7rGminmax _eval_funcrrrr is_integeris_evenis_oddrr is_Rational as_coeff_Mul is_numberr# is_positiver"r@r make_argsrappendrr)rr rrrrrncoefftermscoeffslog_termtermterm_outadd argchangedanewas r2evalexp.evals.8.6 c & &779   ) )qvvs# # ]]aee|uu uu vv  "zz!***vv + A%% %55L S ! !88A;   ) )s377|S\: :  % %>>#& & ZZZ&&r!t,EeG''}} uu  }},!&&.11 !r !&&.00 1&&"QYFz! "6"9Q;//++-LE++QZZ88?? 2 22!&%y((U!&&-@ uu %y,,E!&&1H 000%y,, vv  %wH e,"4(eS))'#(::a=#''MM$'-.68S&\) ?4 ? ZZCCJXX:JJqM1vdC((yy|q( 499Q<0%)  1 JJt$jCyS#Y!??? ;;55L r4c"[R$)z/ Returns the base of the exponential function. )rrr0s r2rexp.base~s vv r4cUS:a[R$US:Xa[R$[U5nU(aUSnUbX1-U- $X-[ U5- $)z: Calculates the next term in the Taylor series expansion. r)rrr?rr)nr{previous_termsps r2 taylor_termexp.taylor_terms\ q566M 655L AJ r"A}uqy tIaL  r4c SSKJnJn URSR 5upVU(a&UR "U40UD6nUR "U40UD6nU"U5U"U5pC[ U5U-[ U5U-4$)a Returns this function as a 2-tuple representing a complex number. Examples ======== >>> from sympy import exp, I >>> from sympy.abc import x >>> exp(x).as_real_imag() (exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x))) >>> exp(1).as_real_imag() (E, 0) >>> exp(I).as_real_imag() (cos(1), sin(1)) >>> exp(1+I).as_real_imag() (E*cos(1), E*sin(1)) See Also ======== sympy.functions.elementary.complexes.re sympy.functions.elementary.complexes.im r)cossin)(sympy.functions.elementary.trigonometricrrrG as_real_imagexpandr.)r1deeprrrr#r"s r2rexp.as_real_imagss0 F1**, 4)5)B4)5)Br7CGSB SWS[))r4c UR(a,[UR[UR5-5nO*U[R LaUR (a[n[U[5(dU[R La'Sn[R"U"U5U"U5U5$U[La.UR (dX RRX5-$[R"XU5$)NcUR(d[U[5(a[UR 5SS06$U$)NrpF)is_Powrr.rrK)rs r2 exp._eval_subs..s7Jq#..q}}??7567r4) r r.r7rrr is_Functionrr _eval_subs_subsr)r1oldnewfs r2rexp._eval_subss ::cggc#((m+,C AFF]sC c3  3!&&=7A>>!D'1S637 7 #:coos00 0""4c22r4cURSR(agURSR(a6[S5*[-URS-[ - nUR $g)NrTr)rGr is_imaginaryrrrrr1arg2s r2rexp._eval_is_extended_realsW 99Q< ( ( YYq\ & &aD519tyy|+b0D<< 'r4cDSn[U"URS55$)Nc3D# URv URv g7fr-) is_complexr])r s r2complex_extended_negative7exp._eval_is_complex..complex_extended_negatives.. ** *s r)rrG)r1rs r2_eval_is_complexexp._eval_is_complexs" +1$))A,?@@r4c UR[- [- R(ag[ URR 5(a@URR (agUR[- R(agggr[)r.rrrerrd is_algebraicr0s r2_eval_is_algebraicexp._eval_is_algebraics^ HHrMA  * * TXX%% & &xx$$((R-,,- 'r4cURR(aURS[RL$URR (a*[ *URS-[- nUR$gr) r.rrGrrkrrrrrs r2_eval_is_extended_positiveexp._eval_is_extended_positives] 88 $ $99QU-[A[R>U-55nU$![0[$4a SnGN/f=f)Nr)signceiling)limitOrderpowsimprlogxrJCannot expand %s around 0c3<># UHn[UT5v M g7fr-)r)rzr r*s r2r|$exp._eval_nseries..s: z#t$$ str3Tr.rcombinecFUR=(a URS;$)N))rq)r{s r2r #exp._eval_nseries.. samm@y0@@r4w) properties)!$sympy.functions.elementary.complexesr*#sympy.functions.elementary.integersr,sympy.series.limitsr-sympy.series.orderr/sympy.simplify.powsimpr1r. _eval_nseriesis_OrderremoveOrrkrr\r anyrGras_leading_termgetnNotImplementedError_taylorsubsr7rrreplacerr)r1r{rr3cdirr,r-r/r1r arg_seriesarg0r7ntermscf exp_seriesrrep simpleratrAr*s @r2rHexp._eval_nseriess >?-,2hh&&qD9   z> !Z'')1a0 1%% %q> ! 1:: K   74@A A : : : :K #J s**18!<AACB "q&QT]FV^^A. Ijooad):; ; $ 0tSVnb 66#;$ H "q&  )A-q1!!tQh-? ?A  )A-q1 1A HHJ AD% 0@ ) - IIammQ&q}}a7G(H I'$Y/ B s1%H00IIc/nSn[U5HPnURXPRSU5nURXS9nUR UR 55 MR [ U6$)Nr)r)rangerrGnseriesrrJr)r1r{rlgrs r2rO exp._taylorsa  qA  IIaL!4A ! !A HHQYY[ !Awr4NcSSKJn URSR5R XS9nUR "US5nU[ RLa[ R$[Xd5(a4[U5[ R:a [U*5$[U5$U[ RLaUR"US5nURSLa [U5$[SU-5e)Nrrr8Fr4)sympy.calculus.utilrrGcancelrLrPrrrr#rr.r-r\r )r1r{r3rRrr rTs r2_eval_as_leading_termexp._eval_as_leading_terms3iil!!#33A3Axx1~ !%%<55L d ( ( $x!&& D5z!t9  155=99Q?D   u $t9 3t<==r4c nSSKJn U"[U-[S- -5[U"[U-5-- $)Nr)rr)rrrr)r1r kwargsrs r2_eval_rewrite_as_sinexp._eval_rewrite_as_sin/s-@1S52a4< 1S3Z<//r4c nSSKJn U"[U-5[U"[U-[S- -5--$)Nr)rr)rrrr)r1r rhrs r2_eval_rewrite_as_cosexp._eval_rewrite_as_cos3s.@1S5zAc!C%"Q$,////r4c HSSKJn SU"US- 5-SU"US- 5- - $)Nr)tanhrJr)%sympy.functions.elementary.hyperbolicro)r1r rhros r2_eval_rewrite_as_tanhexp._eval_rewrite_as_tanh7s(>DQK!d3q5k/22r4c :SSKJnJn UR(aUR"[ [ -5nU(a]UR(aKU"[ U-5U"[ U-5pv[Xd5(d[Xs5(d U[ U--$ggggg)Nr)rr) rrrrrrrrr)r1r rhrrrcosinesines r2_eval_rewrite_as_sqrtexp._eval_rewrite_as_sqrt;suE ::IIbdOE"2e8}c"U(m!&..z47M7M!AdF?*8N.)u r4c :UR(aURVs/sH7n[U[5(dM[ UR5S:XdM5UPM9 nnU(a/[ USRSUR "US55$ggs snfNrJr)rrGrr7lenrr)r1r rhrlogss r2_eval_rewrite_as_Powexp._eval_rewrite_as_PowDsr ::"xxSx!:a+=A#aff+QRBRAxDS47<<?CIId1g,>?? SsBBBrrTrr)rrrrrrr classmethodrrr staticmethodrrrrrr r$r'rHrOrerirlrqrvr|rrr4r2r.r.s45!(ggR   !  !*@ 3 A  -^>*003+@r4r.) metaclasscUR[SS9upUS:XaUR(aX4$UR[5nU(a%UR(aUR(aX4$g)a Try to match expr with $a + Ib$ for real $a$ and $b$. ``match_real_imag`` returns a tuple containing the real and imaginary parts of expr or ``(None, None)`` if direct matching is not possible. Contrary to :func:`~.re()`, :func:`~.im()``, and ``as_real_imag()``, this helper will not force things by returning expressions themselves containing ``re()`` or ``im()`` and it does not expand its argument either. Tas_Addr)NN)as_independentris_realr)exprr_i_s r2match_real_imagrKsX 4 0FB Qw2::x  1 B bjjRZZxr4c\rSrSr%Sr\\\S'\R\R4r SSjr SSjr \SSj5rSr\\S 55rSS jrS rSS jrS rSrSrSrSrSrSrSrSSjrSSjr Sr!g)r7i`ah The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp rGcHUS:XaSURS- $[X5e)z/ Returns the first derivative of the function. rJr)rGr r8s r2r log.fdiffs( q=TYYq\> !$T4 4r4c[$)z3 Returns `e^x`, the inverse function of `\log(x)`. )r.r8s r2r: log.inverser<r4NcVSSKJn SSKJn [ U5nUb[ U5nUS:Xa&US:Xa[ R $[ R$[X!5nU(aU[XU-- 5[U5- -$[U5[U5- $UR(aUR(a[ R$U[ RLa[ R$U[ R La[ R $U[ R"La[ R $U[ R La[ R $UR$(a#UR&S:XaU"UR(5*$UR*(aDUR,[ RLa'UR.R0(a UR.$[3U[.5(a'UR.R0(a UR.$[3U[.5(aUR.R4(af[7UR.5upgU(aGUR8(a6US[:--nU[::a US[:--nU[=U[>-SS9-$O[3U[@5(a[CUR.5$[3X5(aURDRF(a/U"[URD5[URH55$URDR(a*U"[ R"[URH55$[ R $[3X5(aURJ"U5$UR4(anURL(a[:[>-U"U*5-$U[ RLa[ R$U[ RLa[ R$UR(a[ R$URN(dURP"[>5nUbU[ R La[ R $U[ R"La[ R $UR$(aaURR(a'[:[>-[ RT-U"U5-$[:*[>-[ RT-U"U*5-$UR4(Ga+URV(GaURX"[>SS9upURL(a US -nU S -n [=U SS9n U RY[>S S9upgURQ[>5nURZ(GaU(GaURZ(GaURZ(GaoUR(awURF(a)[:[>-[ RT-U"X-5-$URL(a+[:*[>-[ RT-U"X*-5-$gSS K.J/n Xv- Ra5n U *Ra5n [c5n X;aTU "U[eU 5-5nURF(aU"U5[>X--$U"U5[>X[:- --$X;aUU "U[eU 5-5nURF(aU"U5[>X*--$U"U5[>[:X- --$ggggggg![a Of=fU[ RLaU"U5U"U5- $U"U5$) NrrrrJrFrrrT)ratsimp)3rrrrrrrrr&r7 ValueErrorrrrdr?rrrkrrr?r rr.rrrrrrr rrr!rrrrr@rris_nonnegativerr#rrsympy.simplifyrrd_log_atan_tabler$)rr rrrrrrrarg_rr7t1 atan_tablemoduluss r2rlog.evalsx..cl  4=Dqy!855L,,, !+s3q=1CI===s8CI-- =={{(((vv  "zz!***zz!uu SUUaZCEE {" ::#((aff,1I1I77N c3  CGG$<$<77N S ! !cgg&7&7$SWW-FBb&&ad 7!B$JBJrAvE::: Y ' 'cgg& &  ) )ww"""3sww<SWW>>"1#5#5s377|DDuu  % %>>#& & ==AvSD )))))(((uu ;;$$ $zz&&q)E AJJ&::%a000::%&&++!AvU;; "sQw/#uf+== ===S---,,Qu=KE    d/D((4(8FB""1%B}}} rzzz::~~!AvUZ@@ "sQw/#eck2BBB(7(A"B!0!2J")%#d)*;"<>>#&w>#&w[U5nXxR5;a [S UR555n U bX-$GOUR(a+[UR 5[UR"5- $UR$(Ga+/n /n UR Hn U(d"U R&(dU R((akURU 5n[+U[5(a2U R-URU 5R."S 0UD65 MU R-U5 MU R0(aDURU *5nU R-U5 U R-[2R45 MU R-U 5 M [7U 6[[9U 65-$UR:(d[+U[<5(aU(dUR<R>(aWUR@R&(dWUR<S-R&(aUR<S- RB(dUR@R((ajUR@nUR<nURU5n[+U[5(a[EU5UR."S 0UD6-$[EU5U-$OX[+Xt5(aHU(dURFR&(a&U"[URF5/URHQ76$URU5$) Nr)rxrwforceFfactorr)rrrJc3B# UHupU[U5-v M g7fr-r6)rzvalrs r2r|'log._eval_expand_log..>s D)3s8)sr)%sympy.concreterxrwgetrzrGr rB is_Integerr'r(keyssumitemsrr7rr?rrrrr_eval_expand_logr@rrrrr r.rris_nonpositiver!rr)r1rrrxrwrrr rlogargrrnonposr{rrsrts r2rlog._eval_expand_log+s/ '5)8U+  Na dii3$L Liil >>c"AFE~ 3cNffh& D!'') 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"1? 7*:s$j*r4r7c^\rSrSrSr\"\RSSS9*\R4r \ SSj5r SSjr Sr S rS rSS jrSU4S jjrS rSrU=r$)LambertWi\a The Lambert W function $W(z)$ is defined as the inverse function of $w \exp(w)$ [1]_. Explanation =========== In other words, the value of $W(z)$ is such that $z = W(z) \exp(W(z))$ for any complex number $z$. The Lambert W function is a multivalued function with infinitely many branches $W_k(z)$, indexed by $k \in \mathbb{Z}$. Each branch gives a different solution $w$ of the equation $z = w \exp(w)$. The Lambert W function has two partially real branches: the principal branch ($k = 0$) is real for real $z > -1/e$, and the $k = -1$ branch is real for $-1/e < z < 0$. All branches except $k = 0$ have a logarithmic singularity at $z = 0$. Examples ======== >>> from sympy import LambertW >>> LambertW(1.2) 0.635564016364870 >>> LambertW(1.2, -1).n() -1.34747534407696 - 4.41624341514535*I >>> LambertW(-1).is_real False References ========== .. 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