EdiddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZmZmZmZmZmZmZmZddlmZmZmZdd lmZdd lmZmZmZm Z m!Z!dd l"m#Z#dd l$m%Z%dd l&m'Z'ddl(m)Z)m*Z*ddl+m,Z,ddl-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4ddl5m6Z6m7Z7ddl8m9Z9ddl:m;Z;GddeZ<Gdde<Z=GddeZ>GddeZ?dZ@Gdd eZAGd!d"eZBd#S)$)product)Tuple)Expr)sympify)Add)cacheit)FunctionArgumentIndexError expand_log expand_mul FunctionClass PoleErrorexpand_multinomialexpand_complex) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRationalpiI ImaginaryUnit)global_parameters)Pow)S)WildDummy) factorial)arg unpolarifyimreAbs)sqrt) multiplicity perfect_power) factorint)cancelceZdZdZejfZedZddZ dZ edZ dZ dZ d Zd Zd Zd Zd ZdZdZdS)ExpBaseTc|jjSN)expkindselfs Flib/python3.11/site-packages/sympy/functions/elementary/exponential.pyr/z ExpBase.kind)s x}ctS)z= Returns the inverse function of ``exp(x)``. logr1argindexs r2inversezExpBase.inverse-  r3c|j}|j}|s| js|}|r"tj|| fS|tjfS)a- 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) )r. is_negativecould_extract_minus_signrOnefunc)r1r.neg_exps r2as_numer_denomzExpBase.as_numer_denom3sd h/ 51 52244G  *5$))SD//) )QU{r3c|jdS)z7 Returns the exponent of the function. r)argsr0s r2r.z ExpBase.expKs y|r3cH|dt|jfS)z7 Returns the 2-tuple (base, exponent). r4)r@rrDr0s r2 as_base_expzExpBase.as_base_expRsyy||S$)_,,r3cZ||jSr-)r@r.adjointr0s r2 _eval_adjointzExpBase._eval_adjointXs"yy))++,,,r3cZ||jSr-)r@r. conjugater0s r2_eval_conjugatezExpBase._eval_conjugate["yy++--...r3cZ||jSr-)r@r. transposer0s r2_eval_transposezExpBase._eval_transpose^rMr3cX|j}|jr|jrdS|jrdS|jrdSdSNTF)r. is_infiniteis_extended_negativeis_extended_positive is_finiter1r s r2_eval_is_finitezExpBase._eval_is_finiteasLh ? ' t' u = 4  r3c|j|j}|j|jkr1|jj}|rdS|jjrt |rdSdSdS|jSrR)r@rDr.is_zero is_rationalr)r1szs r2_eval_is_rationalzExpBase._eval_is_rationalksv DIty ! 6TY  ! A t" y|| u    = r3c(|jtjuSr-)r.rNegativeInfinityr0s r2 _eval_is_zerozExpBase._eval_is_zerovsx1---r3cz|\}}tjt||d|S)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)rFr _eval_power)r1otherbes r2rezExpBase._eval_powerys:!!1s1a%888%@@@r3c @ddlm}ddlm}jd}|jr,|jr%tjfd|jDSt||r-|jr&| |j g|j RS |S)Nr)Product)Sumc3BK|]}|VdSr-)r@).0xr1s r2 z1ExpBase._eval_expand_power_exp..s-?? ! ??????r3) sympy.concrete.productsrjsympy.concrete.summationsrkrDis_Addis_commutativerfromiter isinstancer@functionlimits)r1hintsrjrkr s` r2_eval_expand_power_expzExpBase._eval_expand_power_exps333333111111il : A#, A<????ch????? ? S ! ! Ac&8 A7499S\22@SZ@@@ @yy~~r3Nr4)__name__ __module__ __qualname__ unbranchedrComplexInfinity_singularitiespropertyr/r:rBr.rFrIrLrPrXr^rareryr3r2r+r+$sJ')N X 0X --- ---////// ! ! !...AAA r3r+c8eZdZdZdZdZdZdZdZdZ dZ d S) 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 TFcPtt|jdSNr)r.r#rDr0s r2 _eval_Abszexp_polar._eval_Abss2dil##$$$r3cBt|jd} |t kp |tk}n#t$rd}YnwxYw|r|St |jd|}|dkr"t|dkrt |S|S)z. Careful! any evalf of polar numbers is flaky rT)r"rDr TypeErrorr. _eval_evalfr#)r1precibadress r2rzexp_polar._eval_evalfs ty|   8%q2vCC   CCC   K$)A,++D11 q5 RWWq[ c77N s4 AAcH||jd|zSr)r@rD)r1rfs r2rezexp_polar._eval_powersyy1e+,,,r3c.|jdjrdSdS)NrT)rDis_extended_realr0s r2_eval_is_extended_realz exp_polar._eval_is_extended_reals" 9Q< ( 4  r3ct|jddkr|tjfSt|Sr)rDrr?r+rFr0s r2rFzexp_polar.as_base_exps5 9Q<1  ; ""4(((r3N) r{r|r}__doc__is_polar is_comparablerrrerrFrr3r2rrsv##JHM%%%   ---)))))r3rceZdZdZdS)ExpMetac|t|jjvrdSt|to|jt juS)NT)r. __class____mro__rurbaserExp1)clsinstances r2__instancecheck__zExpMeta.__instancecheck__s8 ($, , 4(C((DX]af-DDr3N)r{r|r}rrr3r2rrs(EEEEEr3rceZdZdZddZdZedZedZ e e dZ dd Z d Zd Zd Zd ZdZddZdZddZdZdZdZdZdZdS)r.a9 The exponential function, :math:`e^x`. Examples ======== >>> from sympy import exp, I, pi >>> from sympy.abc import x >>> exp(x) exp(x) >>> exp(x).diff(x) exp(x) >>> exp(I*pi) -1 Parameters ========== arg : Expr See Also ======== log r4c2|dkr|St||)z@ Returns the first derivative of this function. r4)r r8s r2fdiffz exp.fdiffs% q= 5K$T844 4r3cddlm}m}|jd}|jr9t jt jz}||| fvr t jS|j t j t jz}|r|| d|zr|| |r t j S|||r t jS|| |t jzr t j S|||t jzrt jSdSdSdSdS)Nr)askQ)sympy.assumptionsrrrDis_MulrrInfinityNaNas_coefficientPiintegerevenr?odd NegativeOneHalf)r1 assumptionsrrr Ioocoeffs r2 _eval_refinezexp._eval_refinesf,,,,,,,,il : //!*,CsSDk! u &C&qtAO';<2233/ . / / / /////r3c8 ddlm}ddlm}ddlm}ddlm}t||r|j Stj rttj|S|jr}|tjur tjS|jr tjS|tjur tjS|tjur tjS|tjur tjSn|tjur tjSt|t.r |jdSt||r0|t|jt|jSt||r|j|S|jr*|jtjtjz}|rd|zj rh|j!r tjS|j"r tj#S|tj$zj!r tj S|tj$zj"r tjSnB|j%r;|dz}|dkr|dz}||kr%||tjztjzS|j&\}}|tjtjfvr|j'r|tjur| }tQ|jr|tjur tjStQ|j)r'tU|tjur tjStQ|j+r tjSdS|gd} } tYj-|D]T} || } t| t.r| | jd} 2dS| j.r| /| RdS| r | tY| zndS|j0rg} g}d}|jD]}|tjur|/|&||}t||rJ|jd|kr#|/|jdd }u|/|| /|| s|rtY| |tc|d zS|jr tjSdS) Nr AccumBounds) MatrixBaseSetExpr logcombinerr4FTrc)2sympy.calculusrsympy.matrices.matricesrsympy.sets.setexprrsympy.simplify.simplifyrrur.r exp_is_powrrr is_NumberrrZr?rr`Zerorr7rDminmax _eval_funcrrrr is_integeris_evenis_oddrr is_Rational as_coeff_Mul is_numberr# is_positiver"r=r make_argsrappendrrr)rr rrrrrncoefftermscoeffslog_termtermterm_outadd argchangedanewas r2evalzexp.evals......666666......666666 c: & &_ @3799   )] @qvs## # ][ @ae| u  u  v  " z!** v   A% %P @5L S ! !N @8A;  [ ) )L @;s37||S\\:: : W % %J @!3>#&& & ZH @&C&qtAO';<???,3+--LE5+QZ88 ?& 22'!&%yy(%U!&-@% u %yy,1E!&1H1 00%yy,& v t %wHF e,, " 4((eS)) $#(:a=#tt' MM$''''44-5?8S&\))4 ? Z @CCJX % %:JJqMMMs1vvdC((%y|q(& 49Q<000%)  1 JJt$$$$ @j @CyS#Y!?!?!??? ; 5L  r3ctjS)z? Returns the base of the exponential function. )rrr0s r2rzexp.base|s v r3c|dkr tjS|dkr tjSt|}|r|d}|||z|z S||zt |z S)zJ Calculates the next term in the Taylor series expansion. r)rrr?rr)nrnprevious_termsps r2 taylor_termzexp.taylor_termsr q5 6M 6 5L AJJ  !r"A !1uqy !tIaLL  r3Tc ddlm}m}|jd\}}|r|j|fi|}|j|fi|}||||}}t ||zt ||zfS)aJ 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.trigonometricrrrD as_real_imagexpandr.)r1deeprxrrr#r"s r2rzexp.as_real_imags0 FEEEEEEE1**,,B  *4))5))B4))5))B3r77CCGGSB SWWS[))r3c|jr*t|jt|jz}n|tjur|jrt}t|ts|tjur+d}tj |||||S|tur%|js||j ||zStj |||S)Ncz|jst|trt|ddin|S)NrdF)is_Powrur.rrF)rs r2z exp._eval_subs..s@7&q#..7#q}}????56r3) rr.r7rrr is_Functionrur _eval_subs_subsr )r1oldnewfs r2rzexp._eval_subss : cgc#(mm+,,CC AF] s C c3   83!&= 877A>!!D''11S66377 7 #: 1co 1sC000 0"4c222r3c|jdjrdS|jdjr?td tjz|jdztjz }|jSdS)NrTr)rDr is_imaginaryrrrrr1arg2s r2rzexp._eval_is_extended_realsa 9Q< ( 4 Yq\ & aDD51?*TYq\9AD@D<   r3cNd}t||jdS)Nc3.K|jV|jVdSr-) is_complexrT)r s r2complex_extended_negativez7exp._eval_is_complex..complex_extended_negatives). * * * * * *r3r)rrD)r1rs r2_eval_is_complexzexp._eval_is_complexs3 + + +11$)A,??@@@r3c|jtjz tjz jrdSt |jjr)|jjrdS|jtjz jrdSdSdSrR)r.rrrr[rrZ is_algebraicr0s r2_eval_is_algebraiczexp._eval_is_algebraicst HqtOao - : 4 TX% & & x$ u(QT/. u     r3c|jjr|jdtjuS|jjr/tj |jdztjz }|jSdSr) r.rrDrr`rrrrrs r2_eval_is_extended_positivezexp._eval_is_extended_positives^ 8 $ 9Q.s2KK#z#m455KKKKKKr3trTr.rcombinec"|jo|jdvS)N))rq)rns r2rz#exp._eval_nseries..sam@y0@r3w) properties) $sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrsympy.series.limitsrsympy.series.orderr sympy.simplify.powsimpr r. _eval_nseriesis_OrderremoveOrr`ranyrDras_leading_termgetnNotImplementedErrorr_taylorsubsr7rrreplacerr)r1rnrrcdirrrr r r arg_seriesarg0rntermscf exp_seriesr simpleratrrs @r2r zexp._eval_nseriess >=====??????------,,,,,,222222h&S&qAD999   "z> !uZ''))1a00 1% % "5Aq>> ! 1:  K KKKKKKK K K K #JJ *s*14888!<<AACCBB#Y/   BBB   #"q& #WQrT]]FVV^^Av.. IIjooad):;; ; 66$A  4 ' H  2"q& 2  T)A-q11!r!tQh-? ?AA  T)A-q11 1A HHJJ GAD% 0 0 0@@ ) - - - IIamQ&q}a7G(H(H I Is.C55D  D cg}d}t|D]b}|||jd|}|||}||ct |S)Nr)r)rangerrDnseriesrr"r)r1rnrlgrs r2r'z exp._taylor sy  q " "A  DIaL!44A !q !!A HHQYY[[ ! ! ! !Awr3Ncddlm}|jd||}|j|d}|t jur t jSt||r""Qss1S5zz\11r3c ddlm}tj}|||z||||ztjdz zzzS)Nr)rr)rrrrr)r1r r<rrs r2_eval_rewrite_as_coszexp._eval_rewrite_as_cos/sO@@@@@@ Os1S5zzAcc!C%!$q&.11111r3c Tddlm}d||dz zd||dz z z S)Nr)tanhr4r)%sympy.functions.elementary.hyperbolicrA)r1r r<rAs r2_eval_rewrite_as_tanhzexp._eval_rewrite_as_tanh4s?>>>>>>DDQKK!dd3q5kk/22r3c Xddlm}m}|jr|jt jt jz}|rk|jrf|t j|z|t j|z}}t||s(t||s|t j|zzSdSdSdSdSdS)Nr)rr) rrrrrrrrrru)r1r r<rrrcosinesines r2_eval_rewrite_as_sqrtzexp._eval_rewrite_as_sqrt8sEEEEEEEE : 9CIad1?233E 9 9"s14:ADJ!&#..9z47M7M9!AOD$888  9 9 9 9 9 99999r3c |jrHd|jD}|r7t|djd|j|dSdSdS)Nclg|]1}t|tt|jdk/|2Srz)rur7lenrD)rmrs r2 z,exp._eval_rewrite_as_Pow..Cs:SSS!:a+=+=S#af++QRBRSASSSr3r)rrDrr)r1r r<logss r2_eval_rewrite_as_Powzexp._eval_rewrite_as_PowAsm : @SSsxSSSD @47<?ICId1g,>,>??? @ @ @ @r3rzTrr)r{r|r}rrr classmethodrrr staticmethodrrrrrrrrr r'r:r=r?rCrGrMrr3r2r.r.s45555///(gg[gRX   ! ! W\ !****@ 3 3 3   AAA    ****X>>>>*222 222 333999@@@@@r3r.) metaclassc|tjd\}}|dkr |jr||fS|tj}|r|jr |jr||fSdS)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_independentrris_realr)exprr_i_s r2match_real_imagr[Hs}  > >FB Qw2:Bx  1? + +B bjRZBx|r3ceZdZUdZeeed<ejej fZ ddZ ddZ e ddZdZeed Zdd Zd Zdd ZdZdZdZdZdZdZdZdZddZddZdS)r7a 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 rDr4cN|dkrd|jdz St||)z? 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A!!S))' $ADIIaLL$A$J$JE$J$JKKKK A]% 1" AKKNNNMM!-0000MM!$$$$:CL 1 11 1 Z ;:c3// ; -1 -sx7K -QTQXYZQZ -"%'!)!; -BE(BS -HGIIaLLa%%-%a==+=1+=+F+F+F+FFF%a==1,, -W % % ; ; 0 ;s3s|,,:sz::::yy~~r3c 2ddlm}m}m}t |jdkr||j|jfi|S|||jdfi|}|dr ||}||d}t||g|dS) Nr)r simplifyinversecombinerr:Tr`measure)key)rr rrrJrDr@r)r1r<r rrrXs r2_eval_simplifyzlog._eval_simplifywsPPPPPPPPPP ty>>Q  =8IDIty1<$''Dz$T***D$>> 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)) rr7Fcomplex)rDrr$rrr rur7)r1rrxsargsarg_abssarg_args r2rzlog.as_real_imags&y|  6&49Q<&t55u55Dt99 t  < t99 99UE " " +$E) (CMM(7777B Bx==(* *r3c|j|j}|j|jkrQ|jddz jrdS|jdjr$t |jddz jrdSdSdS|jSNrr4TF)r@rDrZr[rr1r\s r2r^zlog._eval_is_rationals DIty ! 6TY  ! ! q ) tvay$ DIaL14D3M)N)N u    = r3c|j|j}|j|jkrQ|jddz jrdSt|jddz jr|jdjrdSdSdS|jSr)r@rDrZrrrs r2rzlog._eval_is_algebraics DIty ! 6TY  " ! q ) !tDIaL1,566 !9Q<,! 5 ! !!!> !r3c&|jdjSrrDrUr0s r2rzlog._eval_is_extended_realsy|00r3cl|jd}t|jt|jgSr)rDrrrrZ)r1r]s r2rzlog._eval_is_complexs, IaL!, !)(<(<=>>>r3c<|jd}|jrdS|jSNrF)rDrZrVrWs r2rXzlog._eval_is_finites$il ; 5}r3c,|jddz jSNrr4rr0s r2rzlog._eval_is_extended_positives ! q 66r3c,|jddz jSr)rDrZr0s r2razlog._eval_is_zeros ! q ))r3c,|jddz jSr)rDis_extended_nonnegativer0s r2_eval_is_extended_nonnegativez!log._eval_is_extended_nonnegatives ! q 99r3rc  ddlm}ddlm}|}|st |}|jd|kr|S|jd}t dt d} } |j| || zz} | W| | | | } } | dkrA| |s,| |st | | |zz} | Sd} |j |\} }|j ||z|}nV#tttf$r<|j ||}|jrdz |j ||}|jYnwxYw|r4||r|j|| t j}} n| | |\} }nP#tttf$r6||t j}} YnwxYwt'|| ||zzz dz }|t,r ||}t/||r|| ||\}}|jst | ||zz}|}t5d d d d d d d d d }|jdi|}| sE|r1|| t | jdi|}n.||t |jdi|}||kr|S|||z|zSfd }i}t;j|D]I}| ||\}}||t j|z||<Jt j } i}|}| |zkrkt j!| z | z }|D]1}||t j|||zz||<2|||}| t j z } | |zkkt | ||zz}|D]}|||||zzz }|dkr!|jd"||}| j#r4| j$r-tK|dkr|dtLzt j'zz}|||z|zS)Nrr rkr5cHtjtj}}tj|D]r}||rV|\}}||kr8 ||cS#t$r|tjfcYcSwxYwm||z}s||fSr-) rr?rrrhasrFleadtermrf)rrnrr.rqrs r2 coeff_expz$log._eval_nseries..coeff_exps3E--- $ $::a==$ & 2 2 4 4ID#qy00#'==#3#3333)000#'</////00 VOEE#: s"A99BBr r4rTF) rr7mul power_exp power_base multinomialbasicrprqci}t||D]E\}}||z}|kr5||tj||||zz||<F|Sr-)rrurr)d1d2re1e2exrs r2rzlog._eval_nseries..mulskC!"b// B BB"W6B!ggb!&11BrF2b6MACGJr3rr)(rr rrr7rDrmatchrrr4rfr&rr!r$rrr"r)rr r.rur%rdictr>r(rrrur?rdirrWr=r"rr) r1rnrrr*r r_logxr rr5r0rrrgr\r_dr_reslogflagsrXrptermsrco1rrpkrrs ` r2r zlog._eval_nseriess -,,,,,666666 q66D 9Q<1  KilCyy$s))1 CIa1f    Q41qAAv aeeAhh quuQxx FFQtVO    33<??DAq A14000AA/; 3 3 3 A...A* 3QCKQT222* 3 3  >TXXa[[ >&3&qt444afqAA >yy{{++A..11 3Y? > > >yy{{22155qv1 > 1a1f:> " " ) ) + + 3 3 5 5 55::  1 A a   AyA1} (a&&1T6/CD4Ue e5H4;****D..00 B--// B7tyy$Q007CC(CC5tyys1vv..5AAAAt|  q!tQ' '     M!$$ @ @Dia((GCB//#++--?F2JJ E cAg ]A%%a'E A A!IIb!&11E"R&L@b RB JA cAg !ffqvo % %B 59QW$ $CC 19 -9Q<##At,,D 9  2d88a<  1Q3qt8OCUU1a4^^##s&)C< -1/e$, 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 rFrcNc|tjkr ||S| tj}|jr!|jr tjS|tjur tjS|dtjz kr tjS|t d dz krt d S|dt dzkrt dS|tj dz krtjtjzdz S|tdtjzkr tjS|tj ur tj S|jr tjSt|jr|jr tj S|tjurv|tj dz krtj tjzdz S|dtjz kr tjS|dtdzkrtd SdSdS)Nrrr4r)rrrZrr?rr7rrr.rrr`r)rrnrs r2rz LambertW.evals ; 3q66M  A 9 y v AF{ u BqvI~ %}$SVVGAI~ AwAc!ffH} 1vv QTE!G| .qt+A--CAF OO# v AJ "z!y v QY   *y *))   #QTE!G| #',Q..bi #}$bRj # {"  # #  # #r3r4c8|jd}t|jdkr,|dkr%t||dt|zzz Sn:|jd}|dkr't|||dt||zzz St||)z? Return the first derivative of this function. rr4)rDrJrr )r1r9rnrs r2rzLambertW.fdiffs IaL ty>>Q  ?1} 9{{Aq8A;;$788 9 ! 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