a >> 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_negativeZcould_extract_minus_signrOnefunc)r&r#Zneg_expr'r'r(as_numer_denom0s zExpBase.as_numer_denomcCs |jdS)z7 Returns the exponent of the function. r)argsr%r'r'r(r#Hsz ExpBase.expcCs|dt|jfS)z7 Returns the 2-tuple (base, exponent). r))r1rr3r%r'r'r( as_base_expOszExpBase.as_base_expcCs||jSr")r1r#Zadjointr%r'r'r( _eval_adjointUszExpBase._eval_adjointcCs||jSr")r1r# conjugater%r'r'r(_eval_conjugateXszExpBase._eval_conjugatecCs||jSr")r1r#Z transposer%r'r'r(_eval_transpose[szExpBase._eval_transposecCs.|j}|jr |jrdS|jr dS|jr*dSdSNTF)r# is_infiniteis_extended_negativeis_extended_positive is_finiter&argr'r'r(_eval_is_finite^szExpBase._eval_is_finitecCsH|j|j}|j|jkr>|jj}|r(dS|jjrDt|rDdSn|jSdSr9)r1r3r#is_zero is_rationalr)r&szr'r'r(_eval_is_rationalhs  zExpBase._eval_is_rationalcCs |jtjuSr")r#rNegativeInfinityr%r'r'r( _eval_is_zerosszExpBase._eval_is_zerocCs"|\}}tt||dd|S)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)r4r _eval_power)r&otherber'r'r(rJvs zExpBase._eval_powerc s|ddlm}ddlm}jd}|jrH|jrHtfdd|jDSt ||rr|jrr| |j g|j RS |S)Nr)Product)Sumc3s|]}|VqdSr")r1).0xr%r'r( z1ExpBase._eval_expand_power_exp..) Zsympy.concrete.productsrNZsympy.concrete.summationsrOr3is_AddZis_commutativerZfromiter isinstancer1functionlimits)r&hintsrNrOr?r'r%r(_eval_expand_power_exp|s    zExpBase._eval_expand_power_expN)r))__name__ __module__ __qualname__Z unbranchedrComplexInfinity_singularitiespropertyr$r.r2r#r4r5r7r8r@rErGrJrYr'r'r'r(r!!s"     r!c@s@eZdZdZdZdZddZddZdd Zd d Z d d Z dS) exp_polara; Represent a *polar number* (see g-function Sphinx documentation). Explanation =========== ``exp_polar`` represents the function `Exp: \mathbb{C} \rightarrow \mathcal{S}`, sending the complex number `z = a + bi` to the polar number `r = exp(a), \theta = b`. It is one of the main functions to construct polar numbers. Examples ======== >>> from sympy import exp_polar, pi, I, exp The main difference is that polar numbers don't "wrap around" at `2 \pi`: >>> exp(2*pi*I) 1 >>> exp_polar(2*pi*I) exp_polar(2*I*pi) apart from that they behave mostly like classical complex numbers: >>> exp_polar(2)*exp_polar(3) exp_polar(5) See Also ======== sympy.simplify.powsimp.powsimp polar_lift periodic_argument principal_branch TFcCsddlm}t||jdS)Nr)re)$sympy.functions.elementary.complexesrar#r3)r&rar'r'r( _eval_Abss zexp_polar._eval_AbscCsddlm}m}||jd}z|t kp0|tk}WntyJd}Yn0|rT|St|jd|}|dkr||dkr||S|S)z. Careful! any evalf of polar numbers is flaky rimraT)rbrerar3r TypeErrorr# _eval_evalf)r&precreraibadresr'r'r(rgs  zexp_polar._eval_evalfcCs||jd|SNr)r1r3)r&rKr'r'r(rJszexp_polar._eval_powercCs|jdjrdSdS)NrT)r3is_extended_realr%r'r'r(_eval_is_extended_reals z exp_polar._eval_is_extended_realcCs"|jddkr|tjfSt|Srl)r3rr0r!r4r%r'r'r(r4s zexp_polar.as_base_expN) rZr[r\__doc__is_polar is_comparablercrgrJrnr4r'r'r'r(r`s%r`c@seZdZddZdS)ExpMetacCs&t|jjvrdSt|to$|jtjuS)NT)r# __class____mro__rUrbaserExp1)clsinstancer'r'r(__instancecheck__s zExpMeta.__instancecheck__N)rZr[r\ryr'r'r'r(rrsrrc@seZdZdZd,ddZddZeddZed d Z e e d d Z d-ddZ ddZddZddZddZddZd.ddZddZd/d d!Zd"d#Zd$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 r)cCs|dkr |St||dS)z@ Returns the first derivative of this function. r)N)rr,r'r'r(fdiffsz exp.fdiffcCsddlm}m}|jd}|jrtjtj}||| fvr@tjS| tj tj}|r|| d|r|| |r|tj S|||rtjS|| |tjrtj S|||tjrtjSdS)Nr)askQ)Zsympy.assumptionsr{r|r3is_Mulr ImaginaryUnitInfinityNaNas_coefficientPiintegerZevenr0Zodd NegativeOneHalf)r&Z assumptionsr{r|r?ZIoocoeffr'r'r( _eval_refines"  zexp._eval_refinecCsddlm}ddlm}ddlm}m}ddlm}t ||rF| St j rXt tj|S|jr|tjurntjS|jrztjS|tjurtjS|tjurtjS|tjurtjSn|tjurtjSt |tr|jdSt ||r|t |jt |jSt |tr ||S|jr|tj tj!}|rd|j"r~|j#rDtjS|j$rRtj%S|tj&j#rhtj! S|tj&j$rtj!Sn@|j'r|d}|dkr|d8}||kr||tj tj!S|(\}} |tjtjfvrL| j)rH|tjur| } || jr| tjurtjS|| j*r6|| tjur6tjS|| j+rHtjSdS|gd} } t,-| D]R} || } t | tr| dur| jd} ndSn| j.r| /| ndSqb| r| t,| SdS|j0r|g}g}d}|jD]p}|tjur|/|q||}t ||rJ|jd|kr>|/|jdd }n |/|n |/|q|sd|r|t,||t1|dd S|jrtjSdS) Nr AccumBounds) MatrixBaserd logcombiner}r)FTrH)2sympy.calculusrZsympy.matrices.matricesrrbrerasympy.simplify.simplifyrrUr#rZ exp_is_powrrrv is_NumberrrAr0rrFZeror]r+r3minmaxr _eval_funcr~rrr is_integeris_evenZis_oddrr is_RationalZ as_coeff_Mul is_number is_positiver/r make_argsrqappendrTr)rwr?rrrerarrZncoefftermsZcoeffsZlog_termtermZterm_outaddZ argchangedaZnewar'r'r(evals                                  zexp.evalcCstjS)z? Returns the base of the exponential function. )rrvr%r'r'r(ru{szexp.basecGsT|dkrtjS|dkrtjSt|}|rD|d}|durD|||S||t|S)zJ Calculates the next term in the Taylor series expansion. rN)rrr0rr)nrQprevious_termspr'r'r( 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.trigonometricrrr3 as_real_imagexpandr#)r&deeprXrrrarer'r'r(rszexp.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)NrIF)is_PowrUr#rr4)rr'r'r(s z exp._eval_subs..) rr#r+rurrvZ is_FunctionrUr _eval_subs_subsr)r&oldnewfr'r'r(rszexp._eval_subscCsF|jdjrdS|jdjrBtd tj|jdtj}|jSdS)NrTr})r3rm is_imaginaryrrrrr&Zarg2r'r'r(rns    zexp._eval_is_extended_realcCsdd}t||jdS)Ncss|jV|jVdSr") is_complexr;r?r'r'r(complex_extended_negativesz7exp._eval_is_complex..complex_extended_negativer)rr3)r&rr'r'r(_eval_is_complexszexp._eval_is_complexcCsF|jtjtjjrdSt|jjrB|jjr0dS|jtjjrBdSdSr9)r#rrrrBrrA is_algebraicr%r'r'r(_eval_is_algebraics zexp._eval_is_algebraiccCsB|jjr|jdtjuS|jjr>tj |jdtj}|jSdSrl) r#rmr3rrFrrrrrr'r'r(_eval_is_extended_positives zexp._eval_is_extended_positiverc Csddlm}ddlm}ddlm}ddlm}|j} | j |||d} | j rTd| S|| |d} | t j ur|||||S| t jur|Std} |} z|| j||d |}Wnttfyd}Yn0|r|dkr|||} t| | | }t| || | | }|rF|dkrF||| | ||||d|7}n||| | ||7}|}||d d d }d d}td|gd}|t j|tt j|}|S)Nrceiling)limitOrderpowsimprlogxr)trTr#rcombinecSs|jo|jdvS)N))rq)rQr'r'r(rrSz#exp._eval_nseries..w)Z properties)#sympy.functions.elementary.integersrZsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimprr# _eval_nseriesis_OrderremoveOrrFrras_leading_termgetnNotImplementedErrorr _taylorsubsrrreplacerr)r&rQrrcdirrrrrr?Z arg_seriesarg0rZntermscfZ exp_seriesrZ simpleratrr'r'r(rs@         (zexp._eval_nseriescCsNg}d}t|D]4}|||jd|}|j||d}||qt|S)Nr)r)rangerr3nseriesrrr)r&rQrlgrir'r'r(rs z exp._taylorNcCs\|jdj||d}||d}|tjur:||d}|jdurLt|St d|dS)NrrFCannot expand %s around 0) r3cancelrrrrrr:r#r r&rQrrr?rr'r'r(_eval_as_leading_terms    zexp._eval_as_leading_termcKs8ddlm}tj}|||tjd||||S)Nr)rr})rrrrr)r&r?kwargsrrr'r'r(_eval_rewrite_as_sins zexp._eval_rewrite_as_sincKs8ddlm}tj}|||||||tjdS)Nr)rr})rrrrr)r&r?rrrr'r'r(_eval_rewrite_as_coss zexp._eval_rewrite_as_coscKs,ddlm}d||dd||dS)Nr)tanhr)r})Z%sympy.functions.elementary.hyperbolicr)r&r?rrr'r'r(_eval_rewrite_as_tanh!s zexp._eval_rewrite_as_tanhcKsvddlm}m}|jrr|tjtj}|rr|jrr|tj||tj|}}t ||srt ||sr|tj|SdS)Nr)rr) rrrr~rrrrrrU)r&r?rrrrZcosineZsiner'r'r(_eval_rewrite_as_sqrt%s zexp._eval_rewrite_as_sqrtcKs<|jr8dd|jD}|r8t|djd||dSdS)NcSs(g|] }t|trt|jdkr|qS)r))rUr+lenr3)rPrr'r'r( 0rSz,exp._eval_rewrite_as_Pow..r)r~r3rr)r&r?rZlogsr'r'r(_eval_rewrite_as_Pow.szexp._eval_rewrite_as_Pow)r))T)r)Nr)rZr[r\rorzr classmethodrr_ru staticmethodrrrrrnrrrrrrrrrrrr'r'r'r(r#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. TZas_Addr)NNN)as_independentrris_realr)exprr_i_r'r'r(match_real_imag5s  rc@seZdZUdZeeed<ejej fZ d+ddZ d,ddZ e d-d d Zd d Zeed dZd.ddZddZd/ddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd0d'd(Zd1d)d*ZdS)2r+a The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp r3r)cCs$|dkrd|jdSt||dS)z? Returns the first derivative of the function. r)rN)r3rr,r'r'r(rzpsz log.fdiffcCstS)zC Returns `e^x`, the inverse function of `\log(x)`. )r#r,r'r'r(r.ysz log.inverseNc)Cs$ddlm}ddlm}ddlm}t|}|durt|}|dkrX|dkrRtjStjSzBt ||}|r|t |||t |WSt |t |WSWnt yYn0|tj ur||||S||S|j rJ|jrtjS|tjurtjS|tjurtjS|tjurtjS|tjur*tjS|jrJ|jdkrJ||j S|jrp|jtj urp|jjrp|jStj}t|tr|jjr|jSt|tr|jjrt|j\}} | r`| jr`| dtj;} | tjkr| dtj8} |t | |ddSn^t|t!r||jSt||rJ|j"j#rD|t |j"t |j$SdSnt|t%r`|&|S|jr|j'rtj||| S|tjurtjS|tj urtjS|jrtjS|j(s:|)|} | dur:| tjurtjS| tjurtjS| jr:| j*rtj|tj+|| Stj |tj+|| S|jr |j,r |j-|dd \} } | j'rt| d 9} | d 9} t | dd} | j-|d d \}} | )|} | j.r | r | j.r |j.r |jr| j#rtj|tj+|| | S| j'r tj |tj+|| | Sndd l/m0} | |1} | 1}t2d tjd dtjdt2ddt2dtjdt2dt2dt2ddt2dtjdt2ddt2dtjt3ddt2dt2t2ddd t2dtjt3ddt2d d tjdt2ddtjdt2dt2dt2t2ddtjdt2ddtjt3d dt2t2ddt2dt2dtjt3d dt2ddt2ddtjdt2d t2ddt2t2ddtjdt2ddt2ddtjt3d dt2dt2ddt2dt2dtjt3d ddt2d tjdd t2d dt2d tjddt2d tjt3dddt2d d t2d tjt3ddi}| |vr| | || }|j#r||||| S||||| tjSnR||vr | | || }|j#r||||| S|||tj||SdS)Nr unpolarifyr)Absr)r}FrrrT)ratsimprrr )4rbrrrrrrrr]rr+ ValueErrorrvrrAr0rrrFrrrrrur#rmrrUrrrqrr r`rrrr rr/rTris_nonnegativerrrrZsympy.simplifyrrrr)rwr?rurrrrrrrrZarg_rrt1Z atan_tablemodulusr'r'r(rs                           $    * 0&,,$0$  zlog.evalcCs |tjfS)zE Returns this function in the form (base, exponent). )rr0r%r'r'r(r4szlog.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)`. rrrNr)Tr#rr})rrrrr)rrQrrrr'r'r(rs  zlog.taylor_termTcKsddlm}ddlm}ddlm}m}|dd}|dd}t|j dkrdt |j |j ||d S|j d} | j rt | } d} d } | dur| \} } | | } |r|| } | | vrtd d | D} | dur| | Sn| jrt| jt| jS| jrg} g}| j D]}|s*|js*|jrj| |}t|tr^| | |jfi|n | |n6|jr| | }| ||tjn ||qt| tt|S| j st| t!rT|s | j!j"r| j#js | j!d jr| j!d j$s | j#jr| j#}| j!}| |}t|trF|||jfi|S|||Sn4t| |r|sp| j%jr|t| j%g| j&RS| | S) Nrr) factorint)rOrNforceFfactorr})rrr)css|]\}}|t|VqdSr"r*)rPvalrr'r'r(rR?rSz'log._eval_expand_log..)'rbrZsympy.ntheory.factor_rZsympy.concreterOrNgetrr3r r1Z is_Integerrkeyssumitemsrr+rrr~rrprUr_eval_expand_logr/rrrrrr#rmruis_nonpositiverVrW)r&rrXrrrOrNrr r?rZlogargrrZnonposrQrrLrMr'r'r(r*sl              (    zlog._eval_expand_logcKsddlm}m}m}t|jdkr:||j|jfi|S|||jdfi|}|drf||}||dd}t||g|ddS) Nr)r simplifyinversecombiner}r.TrZmeasure)key)rr rrrr3r1r)r&rr rrrr'r'r(_eval_simplifyes zlog._eval_simplifycKsddlm}m}|jd}|r6|jdj|fi|}||}||krP|tjfS||}|ddrd|d<t|j|fi||fSt||fSdS)a Returns this function as a complex coordinate. Examples ======== >>> from sympy import I, log >>> from sympy.abc import x >>> log(x).as_real_imag() (log(Abs(x)), arg(x)) >>> log(I).as_real_imag() (0, pi/2) >>> log(1 + I).as_real_imag() (log(sqrt(2)), pi/4) >>> log(I*x).as_real_imag() (log(Abs(x)), arg(I*x)) r)rr?r+FcomplexN) rbrr?r3rrrr r+)r&rrXrr?Zsargabsr'r'r(rps   zlog.as_real_imagcCs\|j|j}|j|jkrR|jddjr,dS|jdjrXt|jddjrXdSn|jSdSNrr)TF)r1r3rArBrr&rCr'r'r(rEs   zlog._eval_is_rationalcCs\|j|j}|j|jkrR|jddjr,dSt|jddjrX|jdjrXdSn|jSdSr)r1r3rArrrr'r'r(rs   zlog._eval_is_algebraiccCs |jdjSrlr3r<r%r'r'r(rnszlog._eval_is_extended_realcCs|jd}t|jt|jgSrl)r3rrrrA)r&rDr'r'r(rs zlog._eval_is_complexcCs|jd}|jrdS|jSNrF)r3rAr=r>r'r'r(r@s zlog._eval_is_finitecCs|jddjSNrr)rr%r'r'r(rszlog._eval_is_extended_positivecCs|jddjSr)r3rAr%r'r'r(rGszlog._eval_is_zerocCs|jddjSr)r3Zis_extended_nonnegativer%r'r'r(_eval_is_extended_nonnegativesz!log._eval_is_extended_nonnegativerc s&ddlm}ddlm}ddlm}ddlm}ddlm |sHt |}|j d|krZ|S|j d} t dt d} } | | || } | dur| | | | } } | dkr| |s| |st | | |} | Sd d } z&| |\}}| j|||d }WnNtttfyN| j||d }|jrJd 7| j||d }q&Yn0||\}}|||||d }|tr||}t||r|| ||\}}|jst ||||||Sfd d}i}t|D].}| ||\}}||tj |||<qtj!} i}|}| |krtj"|  | }|D]$}||tj |||||<qZ|||}| tj!7} q8t |||}|D]}|||||7}q|dkr|j d#||}|j$r|j%r||dkr|dt&tj'8}||||S)Nr)re)rrr)productkrc Sstjtj}}t|D]^}||rn|\}}||krvz||WStyj|tjfYS0q||9}q||fSr") rr0rrrhasr4leadtermr)rrQrr#r rur'r'r( coeff_exps    z$log._eval_nseries..coeff_exprr)csNi}||D]:\}}||}|kr||tj||||||<q|Sr")r rr)Zd1Zd2rke1e2exrrr'r(muls $zlog._eval_nseries..mulr})(rbreZsympy.polys.polytoolsrrrrr itertoolsrr+r3rmatchrr rrrr rrrrr#rUrrrrr rrr0rdirrr/rr)r&rQrrrrerrrr?rrrr!rrLrCr_dr&ZptermsrZco1r"rpkrr$rkr'r%r(rsr         "  zlog._eval_nseriesc Csddlm}m}|jd}|j||d}||d}|tjurh|durh|j |d||j r`dndd}|tj tj fvrt d||dkr|tj|S|dkr|||}|jr|j r||j r||d ttjS||S) Nrrd)r-+)r)rr)r})rbrerar3Ztogetherrrrrrr/rFrr r0r)rr1rr) r&rQrrrerarr?Zx0r'r'r(rs   zlog._eval_as_leading_term)r))r))N)T)T)r)Nr) rZr[r\rotTupler__annotations__rrr]r^rzr.rrr4rrrrrrrErrnrr@rrGrrrr'r'r'r(r+Js2 !      ; !   Yr+cs|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 ========== .. [1] https://en.wikipedia.org/wiki/Lambert_W_function rFrHNcCsZ|tjkr||S|dur tj}|jr|jr2tjS|tjurBtjS|dtjkrVtjS|td dkrrtd S|dtdkrtdS|tj dkrtjtjdS|t dtjkrtjS|tj urtj S|jrtjSt |jr|jrtj S|tjurV|tj dkr$tj tjdS|dtjkr:tjS|dt dkrVt d SdS)Nrr}r))rrrArvr0rr+rrr#rrrFr)rwrQrr'r'r(rMsB      z LambertW.evalr)cCsv|jd}t|jdkr:|dkrht||dt|Sn.|jd}|dkrht|||dt||St||dS)z? Return the first derivative of this function. rr)N)r3rr1r)r&r-rQrr'r'r(rzss  zLambertW.fdiffcCs|jd}t|jdkr tj}n |jd}|jrZ|dtjjrDdS|dtjjrdSnb|djr|jr~|dtjjr~dS|js|dtjj rdSn"t |jrt |djr|j rdSdSr) r3rrrrArvrrr/rrrm)r&rQrr'r'r(rns"   zLambertW._eval_is_extended_realcCs |jdjSrl)r3r=r%r'r'r(r@szLambertW._eval_is_finitecCsD|j|j}|j|jkr:t|jdjr@|jdjr@dSn|jSdSr)r1r3rrArrr'r'r(rs   zLambertW._eval_is_algebraicrcCsFt|jdkrB|jd}||d}|js8||S||SdS)Nr)r)rr3rrrAr1rrr'r'r(rs   zLambertW._eval_as_leading_termc st|jdkrddlm}ddlm}|jdj|||dj||d}d}|jrZ|j }|||dkrt fddt d|||D} t | } nt j} | ||||St|||S) Nr)rrrrrcs@g|]8}tj |dt||dt|d|qS)r)r})rr0rr)rPrrr'r(rs  z*LambertW._eval_nseries..)rr3rrrrrZcompute_leading_termrr#rrr rrsuperr) r&rQrrrrrltZlterCrsrr(rs     zLambertW._eval_nseriescCs8|jd}t|jdkr|jSt|j|jdjgSdSr)r3rrAr)r&rQr'r'r(rGs zLambertW._eval_is_zero)N)r))Nr)r)rZr[r\rorrrvr]r^rrrzrnr@rrrrG __classcell__r'r'r5r(r1(s" %  r1N)8typingrr/Zsympy.core.exprrZ sympy.corerZsympy.core.addrZsympy.core.cacherZsympy.core.functionrrr r r r r rZsympy.core.logicrrrZsympy.core.mulrZsympy.core.numbersrrrrZsympy.core.parametersrZsympy.core.powerrZsympy.core.singletonrZsympy.core.symbolrrZ(sympy.functions.combinatorial.factorialsrZ(sympy.functions.elementary.miscellaneousrZ sympy.ntheoryrrZsympy.sets.setexprr r!r`rrr#rr+r1r'r'r'r(s8     (       fK^a