+ ir^RIHt^RIHt^RIHt^RIHt^RIH t ^RI H t H t H t HtHtHtHtHt^RIHtHtHt^RIHt^R IHtHtHtHt^R IHt^R I H!t!^R I"H#t#^R I$H%t%H&t&^RI'H(t(^RI)H*t*^RI+H,t,H-t-H.t.H/t/H0t0^RI1H2t2^RI3H4t4H5t5^RI6H7t7!RR] 4t8!RR]84t9!RR]4t:!RR]8]:R7t;Rt<!RR] 4t=!R R!] 4t>]R"4t?R##)$) annotations)product)Add)cacheit)Expr)DefinedFunctionArgumentIndexError expand_log expand_mul FunctionClass PoleErrorexpand_multinomialexpand_complex) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRationalpiI)global_parameters)Pow)S)WildDummy)sympify) factorial)arg unpolarifyimreAbs)sqrt) multiplicity perfect_power) factorintc]tRt^#tRt]P 3t]R4t RRlt Rt ]R4t Rt RtRtR tR tR tR tR tRtRtR#)ExpBaseTc .VPP#N)expkindselfs&چ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/functions/elementary/exponential.pyr- ExpBase.kind(sxx}}c \#)z- Returns the inverse function of ``exp(x)``. logr/argindexs&&r0inverseExpBase.inverse,  r2c HVP'gV\P3#VPpVPpV'g$V)P'gVP 4pV'd#\PVP V)43#V\P3#)z Returns this with a positive exponent as a 2-tuple (a fraction). Examples ======== >>> from sympy import exp >>> from sympy.abc import x >>> exp(-x).as_numer_denom() (1, exp(x)) >>> exp(x).as_numer_denom() (exp(x), 1) )is_commutativerOner, is_negativecould_extract_minus_signfunc)r/r,neg_exps& r0as_numer_denomExpBase.as_numer_denom2sx """; hh//111224G 55$))SD/) )QUU{r2c (VP^,#)z' Returns the exponent of the function. )argsr.s&r0r, ExpBase.expLs yy|r2c JVP^4\VP!3#)z' Returns the 2-tuple (base, exponent). )r@rrEr.s&r0 as_base_expExpBase.as_base_expSsyy|S$))_,,r2c TVPVPP44#r+)r@r,adjointr.s&r0 _eval_adjointExpBase._eval_adjointYsyy))+,,r2c TVPVPP44#r+)r@r, conjugater.s&r0_eval_conjugateExpBase._eval_conjugate\yy++-..r2c TVPVPP44#r+)r@r, transposer.s&r0_eval_transposeExpBase._eval_transpose_rRr2c VPpVP'd)VP'dR#VP'dR#VP'dR#R#TFN)r, is_infiniteis_extended_negativeis_extended_positive is_finiter/rs& r0_eval_is_finiteExpBase._eval_is_finitebsChh ???'''''' === r2c *VP!VP!pVPVP8XdTVPPpV'dR#VPP'd\ V4'dR#R#R#VP#rX)r@rEr,is_zero is_rationalr)r/szs& r0_eval_is_rationalExpBase._eval_is_rationallsf IItyy ! 66TYY  A"""y||(4"== r2c :VP\PJ#r+)r,rNegativeInfinityr.s&r0 _eval_is_zeroExpBase._eval_is_zerowsxx1----r2c jVP4wr#\P!\W#RR7V4#)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)rHr _eval_power)r/otherbes&& r0rnExpBase._eval_powerzs,!s1%8%@@r2c a^RIHp^RIHpSP^,pVP 'd=VP 'd+\P!V3RlVP44#\WC4'd@VP 'd.V!SPVP4.VPO5!#SPV4#)r)Product)Sumc3F<"TFpSPV4xK R#5ir+)r@).0xr/s& r0 1ExpBase._eval_expand_power_exp..s?h ! hs!) sympy.concrete.productsrtsympy.concrete.summationsrurEis_Addr<rfromiter isinstancer@functionlimits)r/hintsrtrursf, r0_eval_expand_power_expExpBase._eval_expand_power_exps31iil :::#,,,<<?chh?? ?  ! !c&8&8&8499S\\2@SZZ@ @yy~r2N)__name__ __module__ __qualname____firstlineno__ unbranchedrComplexInfinity_singularitiespropertyr-r8rBr,rHrLrPrUr^rerirnr__static_attributes__rr2r0r)r)#ssJ'')N  4 - -// !.A r2r)c@]tRt^tRtRtRtRtRtRt Rt Rt R t R #) 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 TFc L\\VP^,44#r)r,r"rEr.s&r0 _eval_Absexp_polar._eval_Abss2diil#$$r2c J\VP^,4pV\)8*;'g V\8pV'dV#\ VP^,4P V4pV^8d\V4^8d \ V4#V# \dRpLhi;i)z-Careful! any evalf of polar numbers is flaky T)r!rEr TypeErrorr, _eval_evalfr")r/precibadress&& r0rexp_polar._eval_evalfs tyy|  8%%q2vC K$))A,++D1 q5RWq[c7N  C sB B B"!B"c TVPVP^,V,4#r)r@rE)r/ros&&r0rnexp_polar._eval_powersyy1e+,,r2c PVP^,P'dR#R#rTN)rEis_extended_realr.s&r0_eval_is_extended_real exp_polar._eval_is_extended_reals 99Q< ( ( ( )r2c VP^,^8XdV\P3#\P V4#r)rErr=r)rHr.s&r0rHexp_polar.as_base_exps1 99Q<1 ; ""4((r2rN) rrrr__doc__is_polar is_comparablerrrnrrHrrr2r0rrs-#JHM% -)r2rc]tRt^tRtRtR#)ExpMetac \VPP9dR#\V\4;'dVP \ PJ#T)r, __class____mro__rrbaserExp1)clsinstances&&r0__instancecheck__ExpMeta.__instancecheck__s; ($$,, ,(C(DDX]]aff-DDr2rN)rrrrrrrr2r0rrsEr2rca]tRt^tRtRRltRt]R4t] R4t ] ] R44t RRltV3RltR tR tR tR tRR ltRtRtRtRtRtRtRtRtV;t#)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 *V^8XdV#\W4h)z0 Returns the first derivative of this function. )r r6s&&r0fdiff exp.fdiffs q=K$T4 4r2c ^RIHpHpVP^,pVP'EdS\ \ P,pWEV)39d\ P#VP!\\ ,4pV'dV!VP^V,44'dV!VPV44'd\ P#V!VPV44'd\ P#V!VPV\ P ,44'd\ )#V!VPV\ P ,44'd\ #R#R#R#R#)r)askQN)sympy.assumptionsrrrEis_MulrrInfinityNaNas_coefficientrintegerevenr=odd NegativeOneHalf)r/ assumptionsrrrIoocoeffs&& r0 _eval_refineexp._eval_refines,iil :::AJJ,CSDk!uu &&r!t,Eqyy5)**166%=)) uu QUU5\** }},QVVEAFFN344 !r QUU5166>233 4+ r2c ^RIHp^RIHp^RIHp^RIHp\W4'dVP!4#\P'd\\PV4#VP'dV\P Jd\P #VP"'d\P$#V\P$Jd\P#V\P&Jd\P&#V\P(Jd\P*#EMV\P,Jd\P #\V\.4'dVP0^,#\W4'd0V!\VP24\VP444#\W4'dVP6!V4#VP8'EdVP:!\<\>,4pV'Ed ^V,P@'dVPB'd\P$#VPD'd\PF#V\PH,PB'd\>)#V\PH,PD'd\>#MOVPJ'd>V^,pV^8d V^,pWv8wdV!V\<,\>,4#VPL!4wrhV\P(\P&39dVPN'dV\P(JdV)p\QV4P"'d%V\P*Jd\P #\QV4PR'd.\UV4\P*Jd\P,#\QV4PV'd\P*#R#V.Rr\XPZ!V4Fcp V!V 4p \V \.4'dV fV P0^,p K:R#V P\'dV P_V 4KbR# V 'dV \YV !,#R#VP`'d.p .pRpVP0FpV\P$JdVP_V4K*V!V4p\VV4'dRVP0^,V8wd'VP_VP0^,4RpKVP_V4KV P_V4K V 'g V'd \YV !V!\cV!RR7,#VP"'d\P$#R#) r AccumBounds) MatrixBaseSetExpr logcombineNFTrl)2sympy.calculusrsympy.matrices.matrixbasersympy.sets.setexprrsympy.simplify.simplifyrrr,r exp_is_powrrr is_Numberrrar=rrhZerorr5rEminmax _eval_funcrrrr is_integeris_evenis_oddrr is_Rational as_coeff_Mul is_numberr" is_positiver!r>r make_argsrappendr}r)rrrrrrrncoefftermscoeffslog_termtermterm_outadd argchangedanewas&& r0evalexp.evals.8.6 c & &779   ) ) )qvvs# # ]]]aee|uu uu vv  "zz!***vv + A%% %55L S ! !88A;   ) )s377|S\: :  % %>>#& & ZZZ&&r!t,EueG'''}}} uu  }},!&&.111 !r !&&.000 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 ? ZZZCCJXX:JJqM1vdC((yy|q( 499Q<0%)  1 JJt$jCyS#Y!??? ;;;55L r2c "\P#)z/ Returns the base of the exponential function. )rrr.s&r0rexp.base}s vv r2c V^8d\P#V^8Xd\P#\V4pV'dVR,pVeW1,V, #W,\ V4, #)z: Calculates the next term in the Taylor series expansion. )rrr=rr)nrxprevious_termsps&&* r0 taylor_termexp.taylor_terms\ q566M 655L AJ r"A}uqy tIaL  r2c ^RIHpHpVP^,P 4wrVV'd'VP !V3/VBpVP !V3/VBpV!V4V!V4rC\ V4V,\ V4V,3#)a Returns this function as a 2-tuple representing a complex number. Examples ======== >>> from sympy import exp, I >>> from sympy.abc import x >>> exp(x).as_real_imag() (exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x))) >>> exp(1).as_real_imag() (E, 0) >>> exp(I).as_real_imag() (cos(1), sin(1)) >>> exp(1+I).as_real_imag() (E*cos(1), E*sin(1)) See Also ======== sympy.functions.elementary.complexes.re sympy.functions.elementary.complexes.im )cossin)(sympy.functions.elementary.trigonometricrrrE as_real_imagexpandr,)r/deeprrrr"r!s&&, r0rexp.as_real_imagss0 F1**, 4)5)B4)5)Br7CGSB SWS[))r2c <VP'd1\VP\VP4,4pM,V\P JdVP 'd\p\V\4'gV\P Jd(Rp\P!V!V4V!V4V4#V\Jd4VP 'g"W PPW4,#\SV`%W4#)cVP'g\V\4'd\VP 4RR/#T#)rmF)is_Powrr,rrH)rs&r0 exp._eval_subs..s9Jq#..q}}??7567r2) rr,r5rrr is_Functionrr _eval_subs_subssuper)r/oldnewfrs&&& r0r exp._eval_subss :::cggc#((m+,C AFF]sC c3  3!&&=7A>>!D'1S637 7 #:cooos00 0w!#++r2c "VP^,P'dR#VP^,P'dG\^4)\,VP^,,\ , pVP #R#r)rEr is_imaginaryrrrrr/arg2s& r0rexp._eval_is_extended_reals] 99Q< ( ( ( YYq\ & & &aD519tyy|+b0D<< 'r2c LRp\V!VP^,44#)c3F"VPxVPxR#5ir+) is_complexrZ)rs&r0complex_extended_negative7exp._eval_is_complex..complex_extended_negatives.. ** *s!)rrE)r/rs& r0_eval_is_complexexp._eval_is_complexs" +1$))A,?@@r2c JVP\, \, P'dR#\ VPP 4'dJVPP 'dR#VP\, P'dR#R#R#rX)r,rrrbrra is_algebraicr.s&r0_eval_is_algebraicexp._eval_is_algebraicsg HHrMA  * * * TXX%% & &xx$$$((R-,,,- 'r2c (VPP'd$VP^,\PJ#VPP 'd7\ )VP^,,\, pVP#R#)rN) r,rrErrhrrrrrs& r0_eval_is_extended_positiveexp._eval_is_extended_positivesc 88 $ $ $99QV,\A\P>V,44pV# \0\$3d^pELxi;i)r)signceiling)limitOrderpowsimprlogxCannot expand %s around 0c3<<"TFp\VS4xK R#5ir+)r)rwrr's& r0ry$exp._eval_nseries..s: z#t$$ sTFtr0r,rcombinecHVP;'dVPR9#))r9)rq)rxs&r0r #exp._eval_nseries.. samm@@y0@@r2w) properties)!$sympy.functions.elementary.complexesr'#sympy.functions.elementary.integersr)sympy.series.limitsr*sympy.series.orderr,sympy.simplify.powsimpr.r, _eval_nseriesis_OrderremoveOrrhrrYr anyrEras_leading_termgetnNotImplementedError_taylorsubsr5rrreplacerr)r/rxrr0cdirr)r*r,r.r arg_seriesarg0r4ntermscf exp_seriesrrep simpleratr>r's&&&&& @r0rEexp._eval_nseriess1 >?-,2hh&&qD9    z> !Z'')1a0 1%% %q> ! 1:: K    74@A A 3: :333: : : :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 s4(J>>KKc .pRp\V4FWpVPWPP^,V4pVPWR7pVP VP 44KY \ V!#)N)r)rangerrEnseriesrrGr)r/rxrlgrs&&& r0rL exp._taylorsa  qA  IIaL!4A ! !A HHQYY[ !Awr2c ^RIHpVP^,P4P WR7pVP !V^4pV\ PJd\ P#\Wd4'd6\V4\ P8d \V)4#\V4#V\ PJdVP!V^4pVPRJd \V4#\RV,4h)rrr5Fr1)sympy.calculus.utilrrEcancelrIrMrrrr"rr,r*rYr )r/rxr0rOrrrQs&&&& r0_eval_as_leading_termexp._eval_as_leading_terms3iil!!#33A3Axx1~ !%%<55L d ( ( $x!&& D5z!t9  155=99Q?D   u $t9 3t<==r2c ^RIHpV!\V,\^, ,4\V!\V,4,, #)r)r)rrrr)r/rkwargsrs&&, r0_eval_rewrite_as_sinexp._eval_rewrite_as_sin.s-@1S52a4< 1S3Z<//r2c ^RIHpV!\V,4\V!\V,\^, ,4,,#)r)r)rrrr)r/rrers&&, r0_eval_rewrite_as_cosexp._eval_rewrite_as_cos2s.@1S5zAc!C%"Q$,////r2c p^RIHp^V!V^, 4,^V!V^, 4, , #)r)tanh)%sympy.functions.elementary.hyperbolicrl)r/rrerls&&, r0_eval_rewrite_as_tanhexp._eval_rewrite_as_tanh6s(>DQK!d3q5k/22r2c v^RIHpHpVP'dVP!\ \ ,4pV'dtVP'd`V!\ V,4V!\ V,4rv\Wd4'g(\Ws4'gV\ V,,#R#R#R#R#R#)r)rrN) rrrrrrrrr)r/rrerrrcosinesines&&, r0_eval_rewrite_as_sqrtexp._eval_rewrite_as_sqrt:syE :::IIbdOE"2e8}c"U(m!&..z47M7M!AdF?*8N.)u r2c \VP'dVPUu.uF9p\V\4'gK\ VP4^8XgK7VNK; ppV'd<\ V^,P^,VP !V^,44#R#R#uupirN)rrErr5lenrr)r/rrerlogss&&, r0_eval_rewrite_as_Powexp._eval_rewrite_as_PowCsv :::"xxSx!:a+=A#aff+QRBRAAxDS47<<?CIId1g,>?? SsB)B)B)rrrr)rrrrrrr classmethodrrr staticmethodrrrr rrr!r$rErLrbrfrirnrsryr __classcell__rs@r0r,r,s45!(ggR   !  !*@ , A  -^>*003+@@r2r,) metaclasscVP\RR7wrV^8XdVP'dW3#VP\4pV'd(VP'dVP'dW3#R#)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_Add)NN)as_independentris_realr)exprr_i_s& r0match_real_imagrJsa 4 0FB Qw2:::x  1 B bjjjRZZZxr2c]tRtRt$RtR]R&]P]P3t RRlt RRlt ] RRl4t ]]R 44tRR ltR tRR ltR tRtRtRtRtRtRtRtRRltRtRtR#)r5i_ah The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp z tuple[Expr]rEc ZV^8Xd^VP^,, #\W4h)z/ Returns the first derivative of the function. )rEr r6s&&r0r log.fdiffs( q=TYYq\> !$T4 4r2c \#)z3 Returns `e^x`, the inverse function of `\log(x)`. )r,r6s&&r0r8 log.inverser:r2Nc d^RIHp^RIHp\ V4pVe\ V4pV^8Xd(V^8Xd\ P #\ P#\W!4pV'd0V\WV,, 4\V4, ,#\V4\V4, #VP'dVP'd\ P#V\ PJd\ P#V\ P Jd\ P #V\ P"Jd\ P #V\ P Jd\ P #VP$'d%VP&^8XdV!VP(4)#VP*'dGVP,\ PJd)VP.P0'd VP.#\3V\.4'd)VP.P0'd VP.#\3V\.4'dVP.P4'd\7VP.4wrgV'deVP8'dSV^\:,,pV\:8dV^\:,,pV\=V\>,RR7,#EM\3V\@4'd\CVP.4#\3W4'dVPDPF'd0V!\VPD4\VPH44#VPDP'd+V!\ P"\VPH44#\ P #\3W4'dVPJ!V4#VP4'dzVPL'd \:\>,V!V)4,#V\ PJd\ P#V\ PJd\ P#VP'd\ P#VPN'gVPP!\>4pVeV\ P Jd\ P #V\ P"Jd\ P #VP$'d{VPR'd4\:\>,\ PT,V!V4,#\:)\>,\ PT,V!V)4,#VP4'EdVPV'EdVPX!\>RR7wrVPL'dVR ,pV R ,p \=V RR7p V PY\>RR7wrgVPQ\>4pVPZ'EdV'EdVPZ'EdVPZ'EdVP'dVPF'd:\:\>,\ PT,V!W,4,#VPL'd<\:)\>,\ PT,V!W),4,#R#^RI.H/p Wv, Pa4p V )Pa4p \c4p W9dvV !V\eV 4,4pVPF'd!V!V4\>W,,,#V!V4\>W,\:, ,,#W9dwV !V\eV 4,4pVPF'd"V!V4\>W,),,#V!V4\>\:W,, ,,#R#R#R#R#R#R#R# \dMi;iT\ PJdT!T4T!T4, #T!T4#) rrrNFrrT)ratsimpr)3rrrrrrrrr%r5 ValueErrorrrrar=rrrhrrr<rrr,rrrrrrr rrr rrrrr>r}ris_nonnegativerr rrsympy.simplifyrra_log_atan_tabler#)rrrrrrrrrarg_rr4t1 atan_tablemoduluss&&& r0rlog.evals..cl  4=Dqy!855L,,, !+s3q=1CI===s8CI-- ==={{{(((vv  "zz!***zz!uu SUUaZCEE {" :::#((aff,1I1I1I77N c3  CGG$<$<$<77N S ! !cgg&7&7&7$SWW-FBb&&&ad 7!B$JBJrAvE::: Y ' 'cgg& &  ) )ww""""3sww<SWW>>"1#5#5s377|DDuu  % %>>#& & ===AvSD )))))(((uu ;;;$$ $zzz&&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=a>c P^RIHpV^8d\P#\ V4pV^8XdV#V'd6VR,pVe)V!V)V,V,V^,, RRR7#^^V^,,, W^,,,V^,, #)zF Returns the next term in the Taylor series expansion of `\log(1+x)`. r-Tr,r6r)rDr.rrr)rrxrr.rs&&* r0rlog.taylor_terms 3 q566M AJ 6H r"A}ax!|q1u5D%PPAq1uI U+QU33r2c ^RIHpHpVPRR4pVPRR4p\ VP 4^8Xd&\ VP!VP !WR7#VP ^,pVP'd\V4pRp ^p VRJdVwrzVPV4p V'd@\V4pWxP49d!\RVP444p V e W,#EM'VP'd0\VP 4\VP"4, #VP$'Ed9.p .p VP EFp V'g%V P&'gV P('dmVPV 4p\+V\4'd3V P-VPV 4P.!R/VB4KV P-V4KV P0'dEVPV )4pV P-V4V P-\2P44KV P-V 4EK \7V !\\9V !4,#VP:'g\+V\<4'EdV'gVP<P>'dcVP@P&'gcVP<^,P&'d$VP<^, PB'gVP@P('dtVP@pVP<pVPV4p\+V\4'd#\EV4VP.!R/VB,#\EV4V,#M\\+Wt4'dLV'gVPFP&'d(V!\VPF4.VPHO5!#VPV4#) r)rurtforceFfactor)rrNc3J"TFwrV\V4,xK R#5ir+r4)rwvalrs& r0ry'log._eval_expand_log..7s D)3s8)s!#r)%sympy.concreterurtgetrwrEr r@ is_Integerr&r'keyssumitemsrr5rr<rrrrr_eval_expand_logr>rrrrrr,rris_nonpositiver rr)r/rrrurtrrrrlogargrrnonposrxrrprqs&&, r0rlog._eval_expand_log$s/ '5)8U+  Na dii3$L Liil >>>c"AFE~ 3cNffh& D!'') 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A!!S)) DIIaL$A$A$JE$JK A]]] 1" AKKNMM!--0MM!$:CL 11 1 ZZZ:c3//111sxx7K7K7KQTQXQXYZQZQ"%''!)!;!;!;#((BSBSBSHHGGIIaLa%%%a=1+=+=+F+FFF%a=1,,CT % % 0003s||,:szz::yy~r2c J^RIHpHpHp\ VP 4^8Xd"V!VP !VP !3/VB#VP V!VP ^,3/VB4pVR,'d V!V4pV!VRR7p\WP.VR,R7#)r)r simplifyinversecombiner8Trmeasure)key)rr rrrwrEr@r)r/rer rrrs&, r0_eval_simplifylog._eval_simplify]sPP tyy>Q DIItyy1>> from sympy import I, log >>> from sympy.abc import x >>> log(x).as_real_imag() (log(Abs(x)), arg(x)) >>> log(I).as_real_imag() (0, pi/2) >>> log(1 + I).as_real_imag() (log(sqrt(2)), pi/4) >>> log(I*x).as_real_imag() (log(Abs(x)), arg(I*x)) r5Fcomplex)rErr#rrrrr5)r/rrsargsarg_abssarg_args&&, r0rlog.as_real_imaghs&yy| 99Q<&&t5u5Dt9  < t9 99UE " "$E) M((77B Bx=(* *r2c VP!VP!pVPVP8XdVP^,^, P'dR#VP^,P'd8\ VP^,^, P4'dR#R#R#VP#rTFN)r@rErarbrr/rcs& r0relog._eval_is_rationals IItyy ! 66TYY  ! q )))vvay$$$DIIaL14D3M3M)N)N*O$== r2c VP!VP!pVPVP8XdVP^,^, P'dR#\VP^,^, P4'd(VP^,P'dR#R#R#VP#r)r@rErarr rs& r0r!log._eval_is_algebraics IItyy ! 66TYY  ! q )))DIIaL1,556699Q<,,, -7>> !r2c <VP^,P#rrEr[r.s&r0rlog._eval_is_extended_realsyy|000r2c |VP^,p\VP\VP4.4#r)rErrrra)r/rds& r0rlog._eval_is_complexs, IIaL!,, !))(<=>>r2c hVP^,pVP'dR#VP#)rF)rErar\r]s& r0r^log._eval_is_finites%iil ;;;}}r2c JVP^,^, P#rrr.s&r0r$log._eval_is_extended_positives ! q 666r2c JVP^,^, P#r)rErar.s&r0rilog._eval_is_zeros ! q )))r2c JVP^,^, P#r)rEis_extended_nonnegativer.s&r0_eval_is_extended_nonnegative!log._eval_is_extended_nonnegatives ! q 999r2c ` a^RIHp^RIHp^RIHpVP ^,V8XdVf \V4#T#VP ^,pV!RRR7p V^8Xd^pVP!WV ,4p \R4\R4rV PWV ,,4p V eW,W,rV ^8wdxV PV 4'gaV PV 4'gJVfV \V4,MW,p V \V 4V \V4,, , p V #R pV PW^R 7wppYT T,,, ^, P-4P!T ST^R 7pTP\.4'd T!T4p\1TT4'dTP34oT!TT 4wppTf \T4MTpTP4'g\T4T\T4,, TT,,pTpR RRRRRRRRRRRRRRRRR/ pTP6!R/TBpTP94'gETP94'd/TPT)\T4)4P6!R/TBpM+TPT\T44P6!R/TBpTT8XdT#TT!TS,T4,#T3Rlp/p\:P<!TP%44F9pT!TT 4wppTP?T\(P*4T,TT&K; \(P@p /pTpT T,S8d\(PBT ,)T , p TF;p!TP?T!\(P*4T TT!,,,TT!&K= T!TT4pT \(P@, p K\T4T\T4,, TT,,pTF/p!TTT!,P-4T T!,,, pK1 TPD'd\GT 4^8wd^RI$H%p"\MT POT 44F!wp#pTPP'd T#^8XgK!M X#^8dLXPST 4wp pTR\T,\V,T"!\GT 4)^4,, pTPYT, 4pTT!TS,T4,# \\\3dT P!T ST^R 7pTP"'d!S^, oT P!T ST^R 7pK2TP%4PT ^R 7wppEL \d7TP%4P'T ^R 7\(P*ppELi;ii;i)rr+r)rr4Tpositivekr\cd\P\Pr2\P!V4FPpVP V4'd/VP 4wrSWQ8wdVPV4u#KHW$,pKR W#3# \dT\P3uu#i;ir+) rr=rrrhasrHleadtermr)rrxrr,rrs&& r0 coeff_exp$log._eval_nseries..coeff_exps3---::a== & 2 2 4IDy0#'==#33! OE.:  *0#'</0s)B  B/.B/r0rO)rr0rO)rOrr5mulF power_exp power_base multinomialbasicrrc</p\W4FRwr4W4,pVS8gKVPV\P4W,W,,,W%&KT V#r+)rrrr)d1d2re1e2exrs&& r0rlog._eval_nseries..mulsPC!"/W6!ggb!&&1BF26MACG*Jr2 Heavisider),rCr,rrsympy.core.symbolrrEr5rMrmatchrrrrKr rErFrGrIrrrar,rrJrrr?rrrr=rr>r!'sympy.functions.special.delta_functionsr enumeratelseriesras_coeff_exponentrr)$r/rxrr0rOr,rrrr4rdrr\rUrrrprcr_dr_reslogflagsrrptermsrco1rrpkrrrrs$&&f&& r0rElog._eval_nseriess -6+ 99Q<1 !\3q6 3t 3iil # % 19D HHQQ Cy$s)1 GGAdFO =4qAvaeeAhhquuQxx $ Ac!fH!&SVaD k))  F::a:3DAq!Q$Z!^ # # % 3 3AA 3 N 55::1 A a  AA1s1v4}}}a&1SY;&4/CDeT5%ee]E7E7TX%!H;;**D..00--//yy$Q077C(Cyys1v.55AAt| q!tQ' ' MM!))+.Da(GCB/#5F2J/ EE cAg]]A%%a'E!IIb!&&1E"R&L@b RB JA!fqT{"QtV+B 59##%a"g- -C ===RUaZ I$QYYq\24|||qAv31u11!4qr!tBwy"U)Q777hhqD&!U1a4^##S/; FQT:A***QOOAAO> Fyy{++AA+61 Fyy{22112=qvv11 F Fs0U##r!rrrrrrrr)r/rxr0rOrQr4rdcrqrrrrrrrs&&&& r0rblog._eval_as_leading_term'syy|$$& # % 19D IIaa  ::a:3DA 5588qD&!AAv ;t DEEq6M :!qvv+155L11!1? ?!fqT{"s1v4 v  ===RUaZ I$QYYq\2|||qAv31u11!4r!tBwy"U)Q777 7 &&q$&?Cs8O sH)H10H1rrr+rr)rrrrr__annotations__rrrrrr8r{rr|rrrrrrer!rrr^r$rirrErbrrr2r0r5r5_sB ffa//0N5 {L{Lz 4 4 7r 8+@! "1? 7*:r$h*r2r5ca]tRtRtRt]!]PRRR7)]P3t ] RRl4t RRlt Rt RtR tR tRV3R lltR tR tV;t#)LambertWiTa 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 ========== .. [1] https://en.wikipedia.org/wiki/Lambert_W_function Frlc V\P8Xd V!V4#Vf\PpVP'Ed=VP'd\P#V\PJd\P#VR\P, 8Xd\P #V\ ^4)^, 8Xd \ ^4)#V^\ ^4,8Xd \ ^4#V\)^, 8Xd\\,^, #V\^\P,48Xd\P#V\PJd\P#\VP4'd#VP'd\P#V\P Jd~V\)^, 8Xd\)\,^, #VR\P, 8Xd\P #VR\R4,8Xd \^4)#R#R#)Nrr)rrrarr=rr5rrr,rrrhr)rrxrs&&&r0r LambertW.evalysv ;q6M YA 999yyyvv AFF{uu BqvvI~}}$SVGAI~AwAc!fH}1v RCEztAv CAFF O#vv AJJzz! QYY  yyy)))  RCEzr"uQwbi}}$bRj {"! r2c ZVP^,p\VP4^8Xd2V^8Xd*\V4V^\V4,,, #MCVP^,pV^8Xd*\W#4V^\W#4,,, #\W4h)z/ Return the first derivative of this function. )rErwrr )r/r7rxrs&& r0rLambertW.fdiffs IIaL tyy>Q 1}{Aq8A;$788 ! A1}~q!hqn*<'=>> 00r2c xVP^,p\VP4^8Xd\PpMVP^,pVP'dcV^\P , ,P 'dR#V^\P , ,P'dR#R#V^,P'dVP'd1V^\P , ,P 'dR#VP'g/V^\P , ,P'dR#R#\VP4'd;\V^,P4'dVP'dR#R#R#R#r) rErwrrrarrrr>rrr)r/rxrs& r0rLambertW._eval_is_extended_reals IIaL tyy>Q A ! 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