o eƦ@sddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z mZmZmZmZmZmZddlmZmZmZddlmZdd lmZmZmZmZmZdd l m!Z!dd l"m#Z#dd l$m%Z%dd l&m'Z'm(Z(ddl)m*Z*ddl+m,Z,ddl-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4ddl5m6Z6m7Z7ddl8m9Z9Gddde Z:Gddde:Z;GdddeZGdd d e Z?Gd!d"d"e Z@ed#d$ZAd%S)&)product)Tuple)Add)cacheit)Expr)FunctionArgumentIndexError expand_log expand_mul FunctionClass PoleErrorexpand_multinomialexpand_complex) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRationalpiI ImaginaryUnit)global_parameters)Pow)S)WildDummy)sympify) factorial)arg unpolarifyimreAbs)sqrt) multiplicity perfect_power) factorintc@seZdZdZejfZeddZdddZ ddZ ed d Z d d Z d dZ ddZddZddZddZddZddZddZdS)ExpBaseTcCs|jjSN)expkindselfr.VD:\Projects\ConvertPro\env\Lib\site-packages\sympy/functions/elementary/exponential.pyr+(sz ExpBase.kindcCtS)z= Returns the inverse function of ``exp(x)``. logr-argindexr.r.r/inverse,zExpBase.inversecCs@|j}|j}|s| js|}|rtj|| 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*Zneg_expr.r.r/as_numer_denom2s  zExpBase.as_numer_denomcCs |jdS)z7 Returns the exponent of the function. r)argsr,r.r.r/r*Js z ExpBase.expcCs|dt|jfS)z7 Returns the 2-tuple (base, exponent). r0)r;rr=r,r.r.r/ as_base_expQszExpBase.as_base_expcC||jSr))r;r*Zadjointr,r.r.r/ _eval_adjointWzExpBase._eval_adjointcCr?r))r;r* conjugater,r.r.r/_eval_conjugateZrAzExpBase._eval_conjugatecCr?r))r;r*Z transposer,r.r.r/_eval_transpose]rAzExpBase._eval_transposecCs.|j}|jr|jr dS|jrdS|jrdSdSNTF)r* is_infiniteis_extended_negativeis_extended_positive is_finiter-rr.r.r/_eval_is_finite`szExpBase._eval_is_finitecCsJ|j|j}|j|jkr"|jj}|rdS|jjrt|r dSdSdS|jSrE)r;r=r*is_zero is_rationalr)r-szr.r.r/_eval_is_rationaljs  zExpBase._eval_is_rationalcCs |jtjuSr))r*rNegativeInfinityr,r.r.r/ _eval_is_zerou zExpBase._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.r/rVxs zExpBase._eval_powerc s|ddlm}ddlm}jd}|jr$|jr$tfdd|jDSt ||r9|jr9| |j g|j RS |S)Nr)Product)Sumc3s|]}|VqdSr))r;).0xr,r.r/ sz1ExpBase._eval_expand_power_exp..) Zsympy.concrete.productsrZZsympy.concrete.summationsr[r=is_AddZis_commutativerZfromiter isinstancer;functionlimits)r-hintsrZr[rr.r,r/_eval_expand_power_exp~s     zExpBase._eval_expand_power_expNr0)__name__ __module__ __qualname__Z unbranchedrComplexInfinity_singularitiespropertyr+r6r<r*r>r@rCrDrKrPrRrVrdr.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 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.r/ _eval_Absszexp_polar._eval_AbscCsxt|jd}z |t kp|tk}Wn tyd}Ynw|r"|St|jd|}|dkr:t|dkr:t|S|S)z. Careful! any evalf of polar numbers is flaky rT)r!r=r TypeErrorr* _eval_evalfr")r-precibadresr.r.r/rps zexp_polar._eval_evalfcCs||jd|Srm)r;r=)r-rWr.r.r/rVszexp_polar._eval_powercCs|jdjrdSdS)NrT)r=is_extended_realr,r.r.r/_eval_is_extended_reals z exp_polar._eval_is_extended_realcCs"|jddkr |tjfSt|Srm)r=rr:r(r>r,r.r.r/r>s  zexp_polar.as_base_expN) rfrgrh__doc__is_polar is_comparablernrprVrvr>r.r.r.r/rls% rlc@seZdZddZdS)ExpMetacCs&t|jjvrdSt|to|jtjuS)NT)r* __class____mro__r`rbaserExp1)clsinstancer.r.r/__instancecheck__s zExpMeta.__instancecheck__N)rfrgrhrr.r.r.r/rzs rzc@seZdZdZd,ddZddZeddZed d Z e e d d Z d-ddZ ddZddZddZddZddZd.ddZddZd/d d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+ZdS)0r*a9 The exponential function, :math:`e^x`. Examples ======== >>> from sympy import exp, I, pi >>> from sympy.abc import x >>> exp(x) exp(x) >>> exp(x).diff(x) exp(x) >>> exp(I*pi) -1 Parameters ========== arg : Expr See Also ======== log r0cCs|dkr|St||)z@ Returns the first derivative of this function. r0)rr4r.r.r/fdiffs z exp.fdiffcCsddlm}m}|jd}|jr^ttj}||| fvrtjS| t t}|r`|| d|rb|| |r;tj S|||rEtjS|| |tjrRt S|||tjrdtSdSdSdSdS)Nr)askQ)Zsympy.assumptionsrrr=is_MulrrInfinityNaNas_coefficientrintegerZevenr:Zodd NegativeOneHalf)r-Z assumptionsrrrZIoocoeffr.r.r/ _eval_refines*  zexp._eval_refinecCsdddlm}ddlm}ddlm}ddlm}t||r!| St j r*t t j|S|jrU|t jur5t jS|jr;t jS|t jurCt jS|t jurKt jS|t jurSt jSnT|t jur]t jSt|trg|jdSt||rw|t |jt |jSt||r||S|jrO|tt}|rd|j r|j!rt jS|j"rt j#S|t j$j!rt S|t j$j"rtSn|j%r|d}|dkr|d8}||kr||ttS|&\}}|t jt jfvr|j'r|t jur| }t(|jr|t jurt jSt(|j)rt*|t jurt jSt(|j+rt jSdS|gd} } t,-|D](} || } t| tr6| dur3| jd} qdS| j.rA| /| qdS| rM| t,| SdS|j0rg} g}d}|jD]:}|t jurk|/|q\||}t||r|jd|kr|/|jdd }q\|/|q\| /|q\| s|rt,| |t1|dd S|jrt jSdS) Nr AccumBounds) MatrixBaseSetExpr logcombinerr0FTrT)2sympy.calculusrZsympy.matrices.matricesrsympy.sets.setexprrsympy.simplify.simplifyrr`r*rZ exp_is_powrrr~ is_NumberrrLr:rrQZerorir3r=minmax _eval_funcrrrr is_integeris_evenZis_oddrr is_RationalZ as_coeff_Mul is_numberr" is_positiver!r8r make_argsryappendr_r)rrrrrrrZncoefftermsZcoeffsZlog_termtermZterm_outaddZ argchangedaZnewar.r.r/evals                              zexp.evalcCstjS)z? Returns the base of the exponential function. )rr~r,r.r.r/r}{szexp.basecGsT|dkrtjS|dkrtjSt|}|r"|d}|dur"|||S||t|S)zJ Calculates the next term in the Taylor series expansion. rN)rrr:rr)nr]previous_termspr.r.r/ taylor_terms zexp.taylor_termTcKstddlm}m}|jd\}}|r%|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-deeprcrrr"r!r.r.r/rszexp.as_real_imagcCs|jrt|jt|j}n |tjur|jrt}t|ts"|tjur1dd}t |||||S|turA|jsA||j ||St |||S)NcSs&|jst|trt|ddiS|S)NrUF)is_Powr`r*rr>)rr.r.r/s z exp._eval_subs..) rr*r3r}rr~Z is_Functionr`r _eval_subs_subsr)r-oldnewfr.r.r/rszexp._eval_subscCsB|jdjrdS|jdjrtd t|jdt}|jSdS)NrTr)r=ru is_imaginaryrrrrr-Zarg2r.r.r/rvs  zexp._eval_is_extended_realcCsdd}t||jdS)Ncss|jV|jVdSr)) is_complexrGrr.r.r/complex_extended_negatives z7exp._eval_is_complex..complex_extended_negativer)rr=)r-rr.r.r/_eval_is_complexszexp._eval_is_complexcCsD|jttjr dSt|jjr|jjrdS|jtjr dSdSdSrE)r*rrrMrrL is_algebraicr,r.r.r/_eval_is_algebraics  zexp._eval_is_algebraiccCs>|jjr |jdtjuS|jjrt |jdt}|jSdSrm) r*rur=rrQrrrrrr.r.r/_eval_is_extended_positives zexp._eval_is_extended_positiverc sddlmddlm}ddlm}ddlm}ddlm }|j } | j |||d} | j r0d| S|| |d} | tjurD||||S| tjurK|Stfd d | jDrY|Std } |} z|| j||d |}Wn ttfyzd}Ynw|r|dkr|||} t | | | }t | || | | }|dur|t|ini}|||kr|S|r|dkr||| | ||||d|7}n ||| | ||7}|}||d dd}dd}td|gd}|tj|ttj|}|S)Nrsignceiling)limitOrderpowsimprlogxr0c3s|] }t|tfVqdSr))r`r)r\rrr.r/r^sz$exp._eval_nseries..trTr*rcombinecSs|jo|jdvS)N))rq)r]r.r.r/rsz#exp._eval_nseries..w) properties) $sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrZsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimprr* _eval_nseriesis_OrderremoveOrrQranyr=ras_leading_termgetnNotImplementedErrorr _taylorsubsr3rrreplacerr)r-r]rrcdirrrrrrZ arg_seriesarg0rZntermscfZ exp_seriesrrepZ simpleratrr.rr/rsN          (zexp._eval_nseriescCsNg}d}t|D]}|||jd|}|j||d}||qt|S)Nrr)rangerr=nseriesrrr)r-r]rlgrrr.r.r/r s z exp._taylorNcCsddlm}|jdj||d}||d}|tjur tjSt||r5t |tj kr1t | St |S|tjur@| |d}|j durIt |Std|)NrrrFCannot expand %s around 0)Zsympy.calculus.utilrr=cancelrrrrr`r"rr*rrFr )r-r]rrrrrr.r.r/_eval_as_leading_terms         zexp._eval_as_leading_termcKs0ddlm}|t|tdt|t|S)Nr)rr)rrrr)r-rkwargsrr.r.r/_eval_rewrite_as_sin* $zexp._eval_rewrite_as_sincKs0ddlm}|t|t|t|tdS)Nr)rr)rrrr)r-rrrr.r.r/_eval_rewrite_as_cos.rzexp._eval_rewrite_as_coscKs,ddlm}d||dd||dS)Nr)tanhr0r)Z%sympy.functions.elementary.hyperbolicr)r-rrrr.r.r/_eval_rewrite_as_tanh2s  zexp._eval_rewrite_as_tanhcKs|ddlm}m}|jr4|tt}|r6|jr8|t||t|}}t||s:t||s<|t|SdSdSdSdSdS)Nr)rr) rrrrrrrrr`)r-rrrrrZcosineZsiner.r.r/_eval_rewrite_as_sqrt6s  zexp._eval_rewrite_as_sqrtcKs@|jrdd|jD}|rt|djd||dSdSdS)NcSs(g|]}t|trt|jdkr|qSre)r`r3lenr=)r\rr.r.r/ As(z,exp._eval_rewrite_as_Pow..r)rr=rr)r-rrZlogsr.r.r/_eval_rewrite_as_Pow?s zexp._eval_rewrite_as_PowreTrrm)rfrgrhrwrr classmethodrrkr} staticmethodrrrrrvrrrrrrrrrrrr.r.r.r/r*s2   i     -  r*) metaclasscCsN|jtdd\}}|dkr|jr||fS|t}|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. TZas_Addr)NN)as_independentris_realr)exprr_i_r.r.r/match_real_imagFs  r c@seZdZUdZeeed<ejej fZ d+ddZ d+ddZ e d,d d Zd d Zeed dZd-ddZddZd-ddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd.d'd(Zd/d)d*ZdS)0r3a 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=r0cCs |dkr d|jdSt||)z? Returns the first derivative of the function. r0r)r=rr4r.r.r/rs z log.fdiffcCr1)zC Returns `e^x`, the inverse function of `\log(x)`. )r*r4r.r.r/r6r7z log.inverseNcCsddlm}ddlm}t|}|dur`t|}|dkr&|dkr#tjStjSzt||}|r=|t |||t |WSt |t |WSt yNYnw|tj ur\||||S||S|j r|j ritjS|tjurqtjS|tjurytjS|tjurtjS|tjurtjS|jr|jdkr||j S|jr|jtj ur|jjr|jSt|tr|jjr|jSt|tr|jjrt|j\}}|r|jr|dt;}|tkr|dt8}|t|tddSns zlog.as_base_expcGsddlm}|dkr tjSt|}|dkr|S|r1|d}|dur1|| |||ddddSdd |d ||d|dS) zV Returns the next term in the Taylor series expansion of `\log(1+x)`. rrrNr0Tr*rr)rrrrr)rr]rrrr.r.r/rs  $zlog.taylor_termTcKs^ddlm}m}|dd}|dd}t|jdkr&t|j|j||dS|jd}|jrct |}d} d} |durC|\}} ||} |rZt |}|| vrZt d d | D} | durb| | Sn|jrpt|jt|jS|jrg} g} |jD]C} |s| js| jr|| }t|tr| || jd i|qz| |qz| jr|| }| || tjqz| | qzt| tt| S|jst|tr|s|jjr|j js|jdjr|jdj!s|j jr|j }|j}||}t|tr t"||jd i|St"||Snt||r*|s|j#jr*|t|j#g|j$RS||S) Nr)r[rZforceFfactorr)rrr0css |] \}}|t|VqdSr)r2)r\valrr.r.r/r^9sz'log._eval_expand_log..r.)%Zsympy.concreter[rZgetrr=r r;Z is_Integerr&r'keyssumitemsrr3rrrrrxr`r_eval_expand_logr8rrrrrr*rur}is_nonpositiver rarb)r-rrcr[rZrrrrZlogargrrZnonposr]rrXrYr.r.r/r&sp                  zlog._eval_expand_logcKsddlm}m}m}t|jdkr||j|jfi|S|||jdfi|}|dr3||}||dd}t||g|ddS) Nr)r simplifyinversecombinerr6Tr measure)key)rr rrrr=r;r)r-rr rrrr.r.r/_eval_simplify_s zlog._eval_simplifycKs|jd}|r|jdj|fi|}t|}||kr |tjfSt|}|ddr;d|d<t|j|fi||fSt||fS)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)) rr3Fcomplex)r=rr#rrrrr3)r-rrcZsargZsarg_absZsarg_argr.r.r/rjs    zlog.as_real_imagcCs^|j|j}|j|jkr,|jddjrdS|jdjr(t|jddjr*dSdSdS|jSNrr0TF)r;r=rLrMrr-rNr.r.r/rPs   zlog._eval_is_rationalcCs^|j|j}|j|jkr,|jddjrdSt|jddjr(|jdjr*dSdSdS|jSr")r;r=rLrrr#r.r.r/rs   zlog._eval_is_algebraiccC |jdjSrmr=rHr,r.r.r/rvrSzlog._eval_is_extended_realcCs|jd}t|jt|jgSrm)r=rrrrL)r-rOr.r.r/rs zlog._eval_is_complexcCs|jd}|jr dS|jSNrF)r=rLrIrJr.r.r/rKs zlog._eval_is_finitecC|jddjSNrr0r%r,r.r.r/rrAzlog._eval_is_extended_positivecCr'r()r=rLr,r.r.r/rRrAzlog._eval_is_zerocCr'r()r=Zis_extended_nonnegativer,r.r.r/_eval_is_extended_nonnegativerAz!log._eval_is_extended_nonnegativerc$ sddlm}ddlm}ddlm}|jd|kr#|dur!t|S|S|jd}|ddd} |dkr4d}|||| } t d t d } } | | | | } | dur| | | | } } | dkr| | s| | s|durs| t|n| |} | t| | t|7} | Sd d }z | j | |dd \}}WnLt ttfy| j| |dd}|jrd7| j| |dd}|jsz |j | dd\}}Wnt y|j| ddtj}}YnwYnw| || |dj| |dd}| tr||}t||r|||| \}}|durt|n|}|jsyt||t|||}|}dddddddddd }|jdi|}|s[|r[|| t| jdi|}n||t|jdi|}||krp|S||||Sfdd}i}t|D]}||| \}}||tj|||<qtj} i}|}| |krtj |  | } |D]}!||!tj| ||!}|!||!<q|||}| tj7} | |kst||t|||}|D] }!|||!| |!7}q|j"rAt#| dkrAddl$m%}"t&| '| D]\}#}|j(r!|#dkr#nq|#dkrA|)| \} }|dt*t+|"t#|  d7}|| ||}||||S)Nrrr)rrTZpositiver0krc Sstjtj}}t|D]/}||r7|\}}||kr6z||WSty5|tjfYSwq ||9}q ||fSr)) rr:rrrhasr>leadtermr)rr]rr*rr}r.r.r/ coeff_exps    z$log._eval_nseries..coeff_exprr)rrr)rF) rr3mulZ power_expZ power_baseZ multinomialbasicrrcsNi}t||D]\}}||}|kr$||tj||||||<q|Sr))rrrr)Zd1Zd2rte1e2exrr.r/r0s"zlog._eval_nseries..mul Heavisider.),rrrrsympy.core.symbolrr=r3rrmatchr,r-rrr rrrrrrr*r`rrrr9rrrr:rZ nsimplifyr8r!'sympy.functions.special.delta_functionsr6 enumeratelseriesras_coeff_exponentrr)$r-r]rrrrrrrrrOr+rrr.rrXrNr_drtZ_resZlogflagsrr0ZptermsrZco1r2rpkrr4r6rrr.rr/rs      "   "       zlog._eval_nseriescCs|jd}tddd}|dkrd}||||}z |j||dd\}}Wnty<|j|||d} t| YSw||rX||||}|dkrTt d|t|S|t j krl|t j krl|t j j||dSt||t|} |dur~t|n|}| ||7} |j rt|dkrdd lm} t||D] \} } | jr| d krnq| d kr| |\}}| d tt| t| d7} | S) NrrTr*r0r/rrr5r7r8)r=Ztogetherrrr-rrr3r,r rr:rr8r!r;r6r<r=rr>rr)r-r]rrrrrOcrYrrtr6rrrrr?r.r.r/r*s>        zlog._eval_as_leading_termrer)rrrm) rfrgrhrwtTupler__annotations__rrrirjrr6rrr>rrrrr rrPrrvrrKrrRr)rrr.r.r.r/r3[s2  !    }  9     ur3cs|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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