a RG5di@sddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZmZmZmZmZmZmZmZddlmZmZmZdd lmZdd lmZmZmZm Z m!Z!dd l"m#Z#dd l$m%Z%dd l&m'Z'ddl(m)Z)m*Z*ddl+m,Z,ddl-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4ddl5m6Z6m7Z7ddl8m9Z9ddl:m;Z;GdddeZGdddedZ?ddZ@Gd d!d!eZAGd"d#d#eZBd$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)cancelc@seZdZdZejfZeddZdddZ ddZ ed d Z d d Z d dZ ddZddZddZddZddZddZddZdS)ExpBaseTcCs|jjSN)expkindselfr/b/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/functions/elementary/exponential.pyr,)sz ExpBase.kindcCstS)z= Returns the inverse function of ``exp(x)``. logr.argindexr/r/r0inverse-szExpBase.inversecCs@|j}|j}|s | js |}|r6tj|| 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)r.r+neg_expr/r/r0as_numer_denom3s zExpBase.as_numer_denomcCs |jdS)z7 Returns the exponent of the function. r)argsr-r/r/r0r+Ksz ExpBase.expcCs|dt|jfS)z7 Returns the 2-tuple (base, exponent). r1)r:rr=r-r/r/r0 as_base_expRszExpBase.as_base_expcCs||jSr*)r:r+adjointr-r/r/r0 _eval_adjointXszExpBase._eval_adjointcCs||jSr*)r:r+ conjugater-r/r/r0_eval_conjugate[szExpBase._eval_conjugatecCs||jSr*)r:r+ transposer-r/r/r0_eval_transpose^szExpBase._eval_transposecCs.|j}|jr |jrdS|jr dS|jr*dSdSNTF)r+ is_infiniteis_extended_negativeis_extended_positive is_finiter.rr/r/r0_eval_is_finiteaszExpBase._eval_is_finitecCsH|j|j}|j|jkr>|jj}|r(dS|jjrDt|rDdSn|jSdSrE)r:r=r+is_zero is_rationalr)r.szr/r/r0_eval_is_rationalks  zExpBase._eval_is_rationalcCs |jtjuSr*)r+rNegativeInfinityr-r/r/r0 _eval_is_zerovszExpBase._eval_is_zerocCs"|\}}tt||dd|S)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)r>r _eval_power)r.otherber/r/r0rUys zExpBase._eval_powerc s|ddlm}ddlm}jd}|jrH|jrHtfdd|jDSt ||rr|jrr| |j g|j RS |S)Nr)Product)Sumc3s|]}|VqdSr*)r:).0xr-r/r0 z1ExpBase._eval_expand_power_exp..) sympy.concrete.productsrYsympy.concrete.summationsrZr=is_Addis_commutativerfromiter isinstancer:functionlimits)r.hintsrYrZrr/r-r0_eval_expand_power_exps    zExpBase._eval_expand_power_expN)r1)__name__ __module__ __qualname__Z unbranchedrComplexInfinity_singularitiespropertyr,r6r<r+r>r@rBrDrKrPrRrUrhr/r/r/r0r)$s"     r)c@s@eZdZdZdZdZddZddZdd Zd d Z d d Z dS) exp_polara< Represent a *polar number* (see g-function Sphinx documentation). Explanation =========== ``exp_polar`` represents the function `Exp: \mathbb{C} \rightarrow \mathcal{S}`, sending the complex number `z = a + bi` to the polar number `r = exp(a), \theta = b`. It is one of the main functions to construct polar numbers. Examples ======== >>> from sympy import exp_polar, pi, I, exp The main difference is that polar numbers do not "wrap around" at `2 \pi`: >>> exp(2*pi*I) 1 >>> exp_polar(2*pi*I) exp_polar(2*I*pi) apart from that they behave mostly like classical complex numbers: >>> exp_polar(2)*exp_polar(3) exp_polar(5) See Also ======== sympy.simplify.powsimp.powsimp polar_lift periodic_argument principal_branch TFcCstt|jdSNr)r+r"r=r-r/r/r0 _eval_Absszexp_polar._eval_AbscCsxt|jd}z|t kp |tk}Wnty:d}Yn0|rD|St|jd|}|dkrtt|dkrtt|S|S)z. Careful! any evalf of polar numbers is flaky rT)r!r=r TypeErrorr+ _eval_evalfr")r.precibadresr/r/r0rss  zexp_polar._eval_evalfcCs||jd|Srp)r:r=)r.rVr/r/r0rUszexp_polar._eval_powercCs|jdjrdSdS)NrT)r=is_extended_realr-r/r/r0_eval_is_extended_reals z exp_polar._eval_is_extended_realcCs"|jddkr|tjfSt|Srp)r=rr9r)r>r-r/r/r0r>s zexp_polar.as_base_expN) rirjrk__doc__is_polar is_comparablerqrsrUryr>r/r/r/r0ros%roc@seZdZddZdS)ExpMetacCs&t|jjvrdSt|to$|jtjuS)NT)r+ __class____mro__rdrbaserExp1)clsinstancer/r/r0__instancecheck__s zExpMeta.__instancecheck__N)rirjrkrr/r/r/r0r}sr}c@seZdZdZd,ddZddZeddZed d Z e e d d Z d-ddZ ddZddZddZddZddZd.ddZddZd/d d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+ZdS)0r+a9 The exponential function, :math:`e^x`. Examples ======== >>> from sympy import exp, I, pi >>> from sympy.abc import x >>> exp(x) exp(x) >>> exp(x).diff(x) exp(x) >>> exp(I*pi) -1 Parameters ========== arg : Expr See Also ======== log r1cCs|dkr |St||dS)z@ Returns the first derivative of this function. r1N)r r4r/r/r0fdiffsz exp.fdiffcCsddlm}m}|jd}|jrtjtj}||| fvr@tjS| tj tj}|r|| d|r|| |r|tj S|||rtjS|| |tjrtj S|||tjrtjSdS)Nr)askQ)sympy.assumptionsrrr=is_MulrrInfinityNaNas_coefficientPiintegerevenr9odd NegativeOneHalf)r. assumptionsrrrIoocoeffr/r/r0 _eval_refines"  zexp._eval_refinecCsddlm}ddlm}ddlm}ddlm}t||rB| St j rTt t j|S|jr|t jurjt jS|jrvt jS|t jurt jS|t jurt jS|t jurt jSn|t jurt jSt|tr|jdSt||r|t |jt |jSt||r||S|jr|t jt j}|rd|j rz|j!r@t jS|j"rNt j#S|t j$j!rdt j S|t j$j"rt jSn@|j%r|d}|dkr|d8}||kr||t jt jS|&\}}|t jt jfvrH|j'rD|t jur| }t(|jr|t jurt jSt(|j)r2t*|t jur2t jSt(|j+rDt jSdS|gd} } t,-|D]R} || } t| tr| dur| jd} ndSn| j.r| /| ndSq^| r| t,| SdS|j0rxg} g}d}|jD]p}|t jur|/|q||}t||rF|jd|kr:|/|jdd }n |/|n | /|q| s`|rxt,| |t1|dd S|jrt jSdS) Nr AccumBounds) MatrixBaseSetExpr logcombinerr1FTrS)2sympy.calculusrsympy.matrices.matricesrsympy.sets.setexprrsympy.simplify.simplifyrrdr+r exp_is_powrrr is_NumberrrLr9rrQZerorlr3r=minmax _eval_funcrrrr is_integeris_evenis_oddrr is_Rational as_coeff_Mul is_numberr" is_positiver!r7r make_argsr|appendrar)rrrrrrrncoefftermscoeffslog_termtermZterm_outadd argchangedanewar/r/r0evals                                   zexp.evalcCstjS)z? Returns the base of the exponential function. )rrr-r/r/r0r|szexp.basecGsT|dkrtjS|dkrtjSt|}|rD|d}|durD|||S||t|S)zJ Calculates the next term in the Taylor series expansion. rN)rrr9rr)nr\previous_termspr/r/r0 taylor_terms zexp.taylor_termTcKstddlm}m}|jd\}}|rJ|j|fi|}|j|fi|}||||}}t||t||fS)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.trigonometricrrr= as_real_imagexpandr+)r.deeprgrrr"r!r/r/r0rszexp.as_real_imagcCs|jrt|jt|j}n|tjur0|jr0t}t|tsD|tjurbdd}t |||||S|tur|js||j ||St |||S)NcSs&|jst|tr"t|ddiS|S)NrTF)is_Powrdr+rr>)rr/r/r0s z exp._eval_subs..) rr+r3rrr is_Functionrdr _eval_subs_subsr)r.oldnewfr/r/r0rszexp._eval_subscCsF|jdjrdS|jdjrBtd tj|jdtj}|jSdS)NrTr)r=rx is_imaginaryrrrrr.arg2r/r/r0rys    zexp._eval_is_extended_realcCsdd}t||jdS)Ncss|jV|jVdSr*) is_complexrGrr/r/r0complex_extended_negativesz7exp._eval_is_complex..complex_extended_negativer)rr=)r.rr/r/r0_eval_is_complexszexp._eval_is_complexcCsF|jtjtjjrdSt|jjrB|jjr0dS|jtjjrBdSdSrE)r+rrrrMrrL is_algebraicr-r/r/r0_eval_is_algebraics zexp._eval_is_algebraiccCsB|jjr|jdtjuS|jjr>tj |jdtj}|jSdSrp) r+rxr=rrQrrrrrr/r/r0_eval_is_extended_positives zexp._eval_is_extended_positiverc sddlmddlm}ddlm}ddlm}ddlm }|j } | j |||d} | j r`d| S|| |d} | tjur||||S| tjur|Stfd d | jDr|Std } |} z|| j||d |}Wnttfyd}Yn0|r|dkr|||} t | | | }t | || | | }||t||krT|S|r|dkr||| | ||||d|7}n||| | ||7}|}||d dd}dd}td|gd}|tj|ttj|}|S)Nrsignceiling)limitOrderpowsimprlogxr1c3s|]}t|tfVqdSr*)rdr)r[rrr/r0r]r^z$exp._eval_nseries..trTr+rcombinecSs|jo|jdvS)N))rq)r\r/r/r0rr^z#exp._eval_nseries..w) properties) $sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimprr+ _eval_nseriesis_OrderremoveOrrQranyr=ras_leading_termgetnNotImplementedErrorr _taylorsubsr3rrreplacerr)r.r\rrcdirrrrrrZ arg_seriesarg0rntermscf exp_seriesrZ simpleratrr/rr0rsJ         (zexp._eval_nseriescCsNg}d}t|D]4}|||jd|}|j||d}||qt|S)Nrr)rangerr=nseriesrrr)r.r\rlgrur/r/r0r s z exp._taylorNcCsddlm}|jdj||d}||d}|tjur@tjSt||rjt |tj krbt | St |S|tjur| |d}|j durt |Std|dS)NrrrFCannot expand %s around 0)sympy.calculus.utilrr=r(rrrrrdr"rr+rrFr )r.r\rrrrrr/r/r0_eval_as_leading_terms        zexp._eval_as_leading_termcKs8ddlm}tj}|||tjd||||S)Nr)rr)rrrrr)r.rkwargsrrr/r/r0_eval_rewrite_as_sin*s zexp._eval_rewrite_as_sincKs8ddlm}tj}|||||||tjdS)Nr)rr)rrrrr)r.rr rrr/r/r0_eval_rewrite_as_cos/s zexp._eval_rewrite_as_coscKs,ddlm}d||dd||dS)Nr)tanhr1r)%sympy.functions.elementary.hyperbolicr)r.rr rr/r/r0_eval_rewrite_as_tanh4s zexp._eval_rewrite_as_tanhcKsvddlm}m}|jrr|tjtj}|rr|jrr|tj||tj|}}t ||srt ||sr|tj|SdS)Nr)rr) rrrrrrrrrrd)r.rr rrrcosinesiner/r/r0_eval_rewrite_as_sqrt8s zexp._eval_rewrite_as_sqrtcKs<|jr8dd|jD}|r8t|djd||dSdS)NcSs(g|] }t|trt|jdkr|qS)r1)rdr3lenr=)r[rr/r/r0 Cr^z,exp._eval_rewrite_as_Pow..r)rr=rr)r.rr logsr/r/r0_eval_rewrite_as_PowAszexp._eval_rewrite_as_Pow)r1)T)r)Nr)rirjrkrzrr classmethodrrnr staticmethodrrrrryrrrrrr r rrrrr/r/r/r0r+s0  i     ,  r+) metaclasscCsV|jtjdd\}}|dkr*|jr*||fS|tj}|rN|jrN|jrN||fSdSdS)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)NNN)as_independentrris_realr)exprr_i_r/r/r0match_real_imagHs  r#c@seZdZUdZeeed<ejej fZ d+ddZ d,ddZ e d-d d Zd d Zeed dZd.ddZddZd/ddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd0d'd(Zd1d)d*ZdS)2r3a The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp r=r1cCs$|dkrd|jdSt||dS)z? Returns the first derivative of the function. r1rN)r=r r4r/r/r0rsz log.fdiffcCstS)zC Returns `e^x`, the inverse function of `\log(x)`. )r+r4r/r/r0r6sz log.inverseNc)Cs6ddlm}ddlm}t|}|durt|}|dkrL|dkrFtjStjSzBt||}|rz|t |||t |WSt |t |WSWnt yYn0|tj ur||||S||S|j r>|j rtjS|tjurtjS|tjurtjS|tjur tjS|tjurtjS|jr>|jdkr>||j S|jrd|jtj urd|jjrd|jStj}t|tr|jjr|jSt|tr|jjrt|j\}}|rr|jrr|dtj;}|tjkr|dtj8}|t||ddSn|t|t r t!|jSt||r\|j"j#r8|t |j"t |j$S|j"j rT|tjt |j$StjSnt||rr|%|S|jr|j&rtj||| S|tjurtjS|tj urtjS|j rtjS|j'sL|(|} | durL| tjurtjS| tjurtjS| jrL| j)r0tj|tj*|| Stj |tj*|| S|jr2|j+r2|j,|dd\} } | j&r| d 9} | d 9} t| dd} | j,|d d\}}|(|}| j-r2|r2|j-r2|j-r2|j r$|j#rtj|tj*|| |S|j&r2tj |tj*|| | Sndd l.m/} ||0} | 0} t1d tjd dtjd t1ddt1dtjdt1dt1dt1ddt1dtjdt1ddt1dtjt2ddt1dt1t1ddd t1dtjt2ddt1d d tjdt1ddtjdt1dt1dt1t1ddtjdt1ddtjt2d dt1t1ddt1dt1dtjt2d dt1ddt1ddtjdt1d t1ddt1t1ddtjdt1ddt1ddtjt2d dt1dt1ddt1dt1dtjt2d ddt1d tjdd t1d dt1d tjddt1d tjt2dddt1d d t1d tjt2ddi}| |vr| | t3| }|j#r||||| S||||| tjSnR| |vr2| | t3| }|j#r|||||  S|||tj|| SdS)Nrrrr1rFrrrT)ratsimprrr )4rrrrrrrrlr%r3 ValueErrorrrrLr9rrrQrrrrrr+rxrrdrr#r|rr ror rrrrr7raris_nonnegativerrrrsympy.simplifyr%r(r$rr#)rrrrrrrr!r"rarg_r%rt1Z atan_tablemodulusr/r/r0rs                           $    * 0&,,$0$  zlog.evalcCs |tjfS)zE Returns this function in the form (base, exponent). )rr9r-r/r/r0r>&szlog.as_base_expcGsddlm}|dkrtjSt|}|dkr.|S|rb|d}|durb|| |||ddddSdd |d ||d|dS) zV Returns the next term in the Taylor series expansion of `\log(1+x)`. rrrNr1Tr+rr)rrrrr)rr\rrrr/r/r0r,s  zlog.taylor_termTcKsxddlm}m}|dd}|dd}t|jdkrLt|j|j||dS|jd}|jrt |}d} d} |dur|\}} ||} |rt |}|| vrt d d | D} | dur| | Sn|jrt|jt|jS|jrg} g} |jD]} |s| js| jrR|| }t|trF| || jfi|n | |q| jr~|| }| || tjq| | qt| tt| S|jst|tr:|s|jjr|j js|jdjr|jdj!s|j jrn|j }|j}||}t|tr,t"||jfi|St"||Sn4t||rn|sV|j#jrn|t|j#g|j$RS||S) Nr)rZrYforceFfactorr)rr0r1css|]\}}|t|VqdSr*r2)r[valrr/r/r0r]Qr^z'log._eval_expand_log..)%Zsympy.concreterZrYgetrr=r r: is_Integerr&r'keyssumitemsrr3rrrrr{rdr_eval_expand_logr7rrrrrr+rxris_nonpositiver rerf)r.rrgrZrYr0r1rrZlogargrr nonposr\rrWrXr/r/r0r8>sh             (    zlog._eval_expand_logcKsddlm}m}m}t|jdkr:||j|jfi|S|||jdfi|}|drf||}||dd}t||g|ddS) Nr)r simplifyinversecombinerr6Tr$measure)key)rr r;r<rr=r:r)r.r r r;r<r r/r/r0_eval_simplifyws zlog._eval_simplifycKs|jd}|r&|jdj|fi|}t|}||kr@|tjfSt|}|ddrvd|d<t|j|fi||fSt||fSdS)a Returns this function as a complex coordinate. Examples ======== >>> 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)) rr3FcomplexN)r=rr#rrrr3r3)r.rrgZsargZsarg_absZsarg_argr/r/r0rs   zlog.as_real_imagcCs\|j|j}|j|jkrR|jddjr,dS|jdjrXt|jddjrXdSn|jSdSNrr1TF)r:r=rLrMrr.rNr/r/r0rPs   zlog._eval_is_rationalcCs\|j|j}|j|jkrR|jddjr,dSt|jddjrX|jdjrXdSn|jSdSrA)r:r=rLrrrBr/r/r0rs   zlog._eval_is_algebraiccCs |jdjSrpr=rHr-r/r/r0ryszlog._eval_is_extended_realcCs|jd}t|jt|jgSrp)r=rrrrL)r.rOr/r/r0rs zlog._eval_is_complexcCs|jd}|jrdS|jSNrF)r=rLrIrJr/r/r0rKs zlog._eval_is_finitecCs|jddjSNrr1rCr-r/r/r0rszlog._eval_is_extended_positivecCs|jddjSrE)r=rLr-r/r/r0rRszlog._eval_is_zerocCs|jddjSrE)r=is_extended_nonnegativer-r/r/r0_eval_is_extended_nonnegativesz!log._eval_is_extended_nonnegativerc sddlm}ddlm}|}|s(t|}|jd|kr:|S|jd}tdtd} } || || } | dur| | | | } } | dkr| |s| |st| | |} | Sdd} z&| |\} }|j |||d}WnNt t t fy.|j ||d}|jr*d 7|j ||d}qYn0|rZ||rZ|j||d tj} }nHz| |\} }Wn0t t t fy||tj} }Yn0t|| ||d }|tr||}t||r|| ||\}}|jst| ||}|}td d d d d d d d d d }|jfi|}| st|rt|| t| jfi|}n||t|jfi|}||kr|S||||Sfdd}i}t|D].}| ||\}}||tj|||<qtj } i}|}| |krhtj!|  | }|D]$}||tj|||||<q*|||}| tj 7} qt| ||}|D]}|||||7}q||dkr|jd"||}| j#r| j$rt%|dkr|dt&tj'8}||||S)Nrrrkrc Sstjtj}}t|D]^}||rn|\}}||krvz||WStyj|tjfYS0q||9}q||fSr*) rr9rrrhasr>leadtermr*)rr\rr+r1rr/r/r0 coeff_exps    z$log._eval_nseries..coeff_exprr1rTF) rr3mul power_exp power_base multinomialbasicr0r1csNi}t||D]:\}}||}|kr||tj||||||<q|Sr*)rr3rr)d1d2rwe1e2exrr/r0rLs $zlog._eval_nseries..mulr)(rrrrr3r=rmatchrIrJrr*rr rrrrrr(rrr+rdrrdictr8rrrr3r9rdirrr7r!rr) r.r\rrrrr_logxrrHrrrKrrWrNr_drw_reslogflagsr rLZptermsrco1rSrpkrrUr/rr0rs        "  "  zlog._eval_nseriescCs|jd}tdddd}|dkr(d}||||}z|j||dd\}}Wn*tyz|j|||d} t| YS0||r||||}|dkrt d|t|S|t j kr|t j kr|t j j||dSt||t|} |durt|n|}| ||7} |j rt|dkrdd lm} t||D]"\} } | jrZ| d kr@qdq@| d kr| |\}}| d tt j| t| d7} | S) NrrT)realpositiver1)rrr r) Heavisider&)r=togetherrrrJr*rr3rIr rr9rr7r!'sympy.functions.special.delta_functionsrb enumeratelseriesras_coeff_exponentrr)r.r\rrrrrOcrXrrwrbrurrrZr/r/r0r 7s:       "zlog._eval_as_leading_term)r1)r1)N)T)T)r)Nr) rirjrkrztTupler__annotations__rrrlrmrr6rrr>rrrr8r?rrPrryrrKrrRrGrr r/r/r/r0r3]s2 !      9    jr3cs|eZdZdZeejddd ejfZe dddZ dd d Z d d Z d dZ ddZdddZdfdd ZddZZS)LambertWa 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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