o 8Va@s@ddlmZddlmZddlmZddlmZmZm Z m Z m Z m Z ddl mZmZmZddlmZddlmZmZddlmZdd lmZdd lmZdd lmZmZdd lm Z dd l!m"Z"ddl#m$Z$m%Z%GdddeZ&Gddde&Z'Gddde Z(Gddde&e(dZ)ddZ*GdddeZ+GdddeZ,dS))sympify)Add)cacheit)FunctionArgumentIndexError _coeff_isneg expand_mul FunctionClass PoleError) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRational)global_parameters)Pow)S)WildDummy) factorial)sqrt) multiplicity perfect_powerc@seZdZdZejfZeddZdddZ ddZ ed d Z d d Z d dZ ddZddZddZddZddZddZddZdS)ExpBaseTcCs|jjSN)expkindselfr H/usr/lib/python3/dist-packages/sympy/functions/elementary/exponential.pyr#sz ExpBase.kindcCtS)z= Returns the inverse function of ``exp(x)``. logrargindexr r r!inverse'zExpBase.inversecCs@|j}|j}|s| jst|}|rtj|| fS|tjfS)a7 Returns this with a positive exponent as a 2-tuple (a fraction). Examples ======== >>> from sympy.functions 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_negativerrOnefunc)rrZneg_expr r r!as_numer_denom-s  zExpBase.as_numer_denomcCs |jdS)z7 Returns the exponent of the function. r)argsrr r r!rEs z ExpBase.expcCs|dt|jfS)z7 Returns the 2-tuple (base, exponent). r")r,rr.rr r r! as_base_expLszExpBase.as_base_expcC||jSr)r,rZadjointrr r r! _eval_adjointRzExpBase._eval_adjointcCr0r)r,r conjugaterr r r!_eval_conjugateUr2zExpBase._eval_conjugatecCr0r)r,rZ transposerr r r!_eval_transposeXr2zExpBase._eval_transposecCs.|j}|jr|jr dS|jrdS|jrdSdSNTF)r is_infiniteis_extended_negativeis_extended_positive is_finiterargr r r!_eval_is_finite[szExpBase._eval_is_finitecCsJ|j|j}|j|jkr"|jj}|rdS|jjrt|r dSdSdS|jSr6)r,r.ris_zero is_rationalr )rszr r r!_eval_is_rationales  zExpBase._eval_is_rationalcCs |jtjuSr)rrNegativeInfinityrr r r! _eval_is_zerop zExpBase._eval_is_zerocCs"|\}}tt||dd|S)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)r/r _eval_power)rotherber r r!rHss zExpBase._eval_powerc stddlm}m}jd}|jr |jr tfdd|jDSt||r5|jr5| |j g|j RS |S)NrSumProductc3s|]}|VqdSr)r,).0xrr r! }sz1ExpBase._eval_expand_power_exp..) sympyrMrNr.is_AddZis_commutativerZfromiter isinstancer,functionlimits)rhintsrMrNr<r rr!_eval_expand_power_expys   zExpBase._eval_expand_power_expNr")__name__ __module__ __qualname__Z unbranchedrComplexInfinity_singularitiespropertyrr(r-rr/r1r4r5r=rBrDrHrXr r r r!rs$      rc@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)$sympy.functions.elementary.complexesrarr.)rrar r r! _eval_Abss zexp_polar._eval_AbscCsddlm}m}m}||jd}z || kp||k}Wn ty'd}Ynw|r,|St|jd|}|dkrD||dkrD||S|S)z. Careful! any evalf of polar numbers is flaky r)impiraT)rRrdrerar. TypeErrorr _eval_evalf)rZprecrdreraiZbadresr r r!rgs zexp_polar._eval_evalfcCs||jd|SNr)r,r.)rrIr r r!rHszexp_polar._eval_powercCs|jdjrdSdS)NrT)r.is_extended_realrr r r!_eval_is_extended_reals z exp_polar._eval_is_extended_realcCs"|jddkr |tjfSt|Srj)r.rr+rr/rr r r!r/s  zexp_polar.as_base_expN) rZr[r\__doc__is_polar is_comparablercrgrHrlr/r r r r!r`s% r`c@seZdZddZdS)ExpMetacCs&t|jjvrdSt|to|jtjuS)NT)r __class____mro__rTrbaserExp1)clsinstancer r r!__instancecheck__s zExpMeta.__instancecheck__N)rZr[r\rwr r r r!rps rpc@seZdZdZd,ddZddZeddZed d Z e e d d Z d-ddZ ddZddZddZddZddZd.ddZddZd/d d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+ZdS)0ra\ The exponential function, :math:`e^x`. Examples ======== >>> from sympy.functions import exp >>> from sympy.abc import x >>> from sympy import I, pi >>> exp(x) exp(x) >>> exp(x).diff(x) exp(x) >>> exp(I*pi) -1 Parameters ========== arg : Expr See Also ======== log r"cCs|dkr|St||)z@ Returns the first derivative of this function. r")rr&r r r!fdiffs z exp.fdiffcCsddlm}m}|jd}|jrctjtj}||| fvr tjS| tj tj}|re|| d|rg|| |r>tj S|||rHtjS|| |tjrVtj S|||tjritjSdSdSdSdS)Nr)askQ)Zsympy.assumptionsryrzr.is_Mulr ImaginaryUnitInfinityNaNas_coefficientPiZintegerZevenr+Zodd NegativeOneHalf)rZ assumptionsryrzr<ZIoocoeffr r r! _eval_refines*  zexp._eval_refinecCsxddlm}ddlm}ddlm}ddlm}m}m }t ||r%| St j r.ttj|S|jrY|tjur9tjS|jr?tjS|tjurGtjS|tjurOtjS|tjurWtjSnZ|tjuratjSt |trk|jdSt ||r{|t |jt |jSt ||r||S|jrY|tj tj!}|rd|j"r|j#rtjS|j$rtj%S|tj&j#rtj! S|tj&j$rtj!Sn|j'r|d} | dkr| d8} | |kr|| tj tj!S|(\}} |tjtjfvr| j)r|tjur| } || jr| tjurtjS|| j*r|| tjurtjS|| j+rtjSdS|gd} } t,-| D](} || }t |tr@| dur=|jd} q%dS| j.rK| /| q%dS| rW| t,| SdS|j0rg}g}d}|jD]:}|tjuru|/|qf||}t ||r|jd|kr|/|jdd }qf|/|qf|/|qf|s|rt,||t1|dd S|jrtjSdS) Nr AccumBoundsSetExpr) MatrixBase)rd logcombinerar{r"FTrF)2sympy.calculusrsympy.sets.setexprrZsympy.matrices.matricesrrRrdrrarTrrZ exp_is_powrrrt is_Numberrr>r+r~rCZeror]r%r.minmax _eval_funcr|rrr} is_integeris_evenZis_oddrr is_RationalZ as_coeff_Mul is_number is_positiver*r make_argsroappendrSr)rur<rrrrdrrarZncoefftermsZcoeffsZlog_termtermZterm_outaddZ argchangedaZnewar r r!evals                             zexp.evalcCstjS)z? Returns the base of the exponential function. )rrtrr r r!rsxszexp.basecGsT|dkrtjS|dkrtjSt|}|r"|d}|dur"|||S||t|S)zJ Calculates the next term in the Taylor series expansion. rN)rrr+rr)nrPprevious_termspr r r! taylor_terms zexp.taylor_termTcKstddlm}m}|jd\}}|r%|j|fi|}|j|fi|}||||}}t||t||fS)aq Returns this function as a 2-tuple representing a complex number. Examples ======== >>> from sympy import I >>> from sympy.abc import x >>> from sympy.functions import exp >>> 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)rdeeprWrrrardr 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)NrGF)is_PowrTrrr/)rr r r!s z exp._eval_subs..) rrr%rsrrtZ is_FunctionrTr _eval_subsZ_subsr)roldnewfr r r!rszexp._eval_subscCsF|jdjrdS|jdjr!td tj|jdtj}|jSdS)NrTr{)r.rk is_imaginaryrr}rrrZarg2r r r!rls   zexp._eval_is_extended_realcCsdd}t||jdS)Ncss|jV|jVdSr) is_complexr8r<r r r!complex_extended_negatives z7exp._eval_is_complex..complex_extended_negativer)r r.)rrr r r!_eval_is_complexszexp._eval_is_complexcCsJ|jtjtjjr dSt|jjr!|jjrdS|jtjjr#dSdSdSr6)rrrr}r?r r> is_algebraicrr r r!_eval_is_algebraics zexp._eval_is_algebraiccCsB|jjr |jdtjuS|jjrtj |jdtj}|jSdSrj) rrkr.rrCrr}rrrr r r!_eval_is_extended_positives zexp._eval_is_extended_positiverc Csddlm}m}m}m}m} m} |j} | j|||d} | j r"d| S|| |d} | t j ur6||||S| t j ur=|Std}|}z|| j||d|}Wn ttfy^d}Ynw|rk|dkrk|||}t|||}t| ||| | }|r|dkr||| | ||||d|7}n ||| | ||7}|}||ddd }d d }| d |gd }|d|| d|}|S)Nr)ceilinglimitOrderpowsimprexpand_complexrlogxr"trTrrZcombinecSs|jo|jdvS)N))rq)rPr r r!rsz#exp._eval_nseries..w)Z propertiesr)rRrrrrrrr _eval_nseriesis_OrderremoveOrrCr~ras_leading_termgetnNotImplementedErrorr _taylorsubsrreplace)rrPrrcdirrrrrrrr<Z arg_seriesarg0rZntermsZcfZ exp_seriesrZ simpleratrr r r!rs<      (zexp._eval_nseriescCsNg}d}t|D]}|||jd|}|j||d}||qt|S)Nr)r)rangerr.nseriesrrr)rrPrlgrhr r r!rs z exp._taylorNcCsX|jdj||d}||d}|tjur||d}|jdur&t|St d|)NrrFCannot expand %s around 0) r.cancelrrrrrr7rr rrPrrr<rr r r!_eval_as_leading_term s     zexp._eval_as_leading_termcKs8ddlm}tj}|||tjd||||S)Nr)rr{)rRrrr}r)rr<kwargsrIr r r!_eval_rewrite_as_sin &zexp._eval_rewrite_as_sincKs8ddlm}tj}|||||||tjdS)Nr)rr{)rRrrr}r)rr<rrrr r r!_eval_rewrite_as_cosrzexp._eval_rewrite_as_coscKs,ddlm}d||dd||dS)Nr)tanhr"r{)rRr)rr<rrr r r!_eval_rewrite_as_tanhs  zexp._eval_rewrite_as_tanhcKsddlm}m}|jr9|tjtj}|r;|jr=|tj||tj|}}t ||s?t ||sA|tj|SdSdSdSdSdS)Nr)rr) rrrr|rrrr}rrT)rr<rrrrZcosineZsiner r r!_eval_rewrite_as_sqrt s zexp._eval_rewrite_as_sqrtcKs@|jrdd|jD}|rt|djd||dSdSdS)NcSs(g|]}t|trt|jdkr|qSrY)rTr%lenr.)rOrr r r! +s(z,exp._eval_rewrite_as_Pow..r)r|r.rr)rr<rZlogsr r r!_eval_rewrite_as_Pow)s zexp._eval_rewrite_as_PowrYTrrj)rZr[r\rmrxr classmethodrr_rs staticmethodrrrrrlrrrrrrrrrrrr r r r!rs2   i   ! #   r) metaclasscCsR|jtjdd\}}|dkr|jr||fS|tj}|r'|jr'|jr'||fSdS)a Try to match expr with a + b*I 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 ``re()``, ``im()``, ``as_real_imag()``, this helper won't force things by returning expressions themselves containing ``re()`` or ``im()`` and it doesn't expand its argument either. TZas_Addr)NN)as_independentrr}is_realr)exprr_i_r r r!match_real_imag0s  rc@seZdZdZejejfZd*ddZd*ddZ e d+dd Z d d Z e ed d Zd,ddZddZd,ddZddZddZddZddZddZdd Zd!d"Zd#d$Zd-d&d'Zd.d(d)ZdS)/r%a The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp r"cCs |dkr d|jdSt||)z? Returns the first derivative of the function. r"r)r.rr&r r r!rxhs z log.fdiffcCr#)zC Returns `e^x`, the inverse function of `\log(x)`. )rr&r r r!r(qr)z log.inverseNcCsDddlm}ddlm}ddlm}ddlm}t|}|durlt|}|dkr2|dkr/t j St j Szt ||}|rI|t |||t |WSt |t |WStyZYnw|t jurh||||S||S|jr|jrut j S|t jur}t jS|t jurt jS|t jurt jS|t j urt j S|jr|jdkr||j S|jr|jt jur|jjr|jSt j}t|tr|jjr|jSt|tr|jjrt |j\} } | r| j!r| dt j";} | t j"kr| dt j"8} | t#| |dd Sn-t|t$r||jSt||r|j%j&r|t |j%t |j'SdSt||r"|(|S|jrF|j)r4t j"||| S|t j ur=t j S|t jurFt jS|jrMt j S|j*s|+|} | dur| t jurdt jS| t jurmt jS| jr| j,rt j"|t j-|| St j" |t j-|| S|jr|j.r|j/|dd \} } | j)r| d 9} | d 9} t#| dd } | j/|d d \} } | +|} | j0r| r| j0r| j0r| jr| j&rt j"|t j-|| | S| j)rt j" |t j-|| | SdSdd l1m2} | | 3}| 3}it4dt j"ddt j"dt4ddt4dt j"dt4dt4dt4ddt4dt j"dt4ddt4dt j"t5ddt4dt4t4ddd t4dt j"t5ddt4ddt j"dt4ddt j"dt4dt4dt4t4ddt j"dt4ddt j"t5ddt4t4ddt4dt4dt j"t5ddt4ddt4ddt j"dt4d t4ddt4t4ddt j"dt4ddt4ddt j"t5ddt4dt4ddt4dt4dt j"t5dddt4dt j"dd t4ddt4dt j"ddt4dt j"t5dddt4dd t4dt j"t5ddi}||vrk| | || }| j&r^|||||S|||||t j"S||vr| | || }| j&r||||| S|||t j"||SdSdSdSdSdSdSdS)Nr) unpolarifyrr)Absr"r{FrrrT)ratsimprrr )6rRrrrrrrbrrrrr]rr% ValueErrorrtrr>r+rr~rCrrrrrsrrkr}rTrrrorrr`rrrrr*rSris_nonnegativerrrrZsympy.simplifyrrrr)rur<rsrrrrrrrrrZarg_rrt1Z atan_tablemodulusr r r!rws"                                 * 0 &  ,  ,$0$   'zlog.evalcCs |tjfS)zE Returns this function in the form (base, exponent). )rr+rr r r!r/ s 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)`. r)rrNr"Trrr{)rRrrrr)rrPrrrr r r!rs  $zlog.taylor_termTcKsrddlm}m}m}ddlm}m}|dd}|dd} t|j dkr0||j |j ||dS|j d} | j rmt | } d} d } | durM| \} } | | } | rd|| } | | vrdtd d | D} | durl| | Sn| jrzt| jt| jS| jrg}g}| j D]C}|s|js|jr| |}t|tr|| |jd i|q||q|jr| | }|||tjq||qt|tt|S| jst| t r|s| j j!r| j"js| j d jr| j d j#s| j"jr| j"}| j }| |}t|tr|||jd i|S|||Snt| |r4|s(| j$jr4|t| j$g| j%RS| | S) Nr)r expand_log factorintrLforceFfactorr{)rrr"css |] \}}|t|VqdSrr$)rOvalrr r r!rQ7sz'log._eval_expand_log..r )&rRrrrZsympy.concreterMrNgetrr.r,Z is_Integerrkeyssumitemsrr%rrr|rrnrTr_eval_expand_logr*rrrrrrrkrsis_nonpositiverUrV)rrrWrrrrMrNrrr<rZlogargrrZnonposrPrrJrKr r r!r #sr                  zlog._eval_expand_logcKsddlm}m}m}t|jdkr||j|jfi|S|||jdfi|}|dr3||}||dd}t||g|ddS) Nr)rsimplifyinversecombiner{r(TrZmeasure)key)Zsympy.simplify.simplifyrr r rr.r,r)rrrr r rr r r!_eval_simplify]s zlog._eval_simplifycKsddlm}m}|jd}|r|jdj|fi|}||}||kr(|tjfS||}|ddrCd|d<t|j|fi||fSt||fS)a Returns this function as a complex coordinate. Examples ======== >>> from sympy import I >>> from sympy.abc import x >>> from sympy.functions import log >>> 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%Fcomplex) rRrr<r.rrrrr%)rrrWrr<Zsargabsr r r!rhs    zlog.as_real_imagcCs^|j|j}|j|jkr,|jddjrdS|jdjr(t|jddjr*dSdSdS|jSNrr"TF)r,r.r>r?r rr@r r r!rBs   zlog._eval_is_rationalcCs^|j|j}|j|jkr,|jddjrdSt|jddjr(|jdjr*dSdSdS|jSr)r,r.r>r rrr r r!rs   zlog._eval_is_algebraiccC |jdjSrjr.r9rr r r!rlrEzlog._eval_is_extended_realcCs|jd}t|jt|jgSrj)r.r rr r>)rrAr r r!rs zlog._eval_is_complexcCs|jd}|jr dS|jSNrF)r.r>r:r;r r r!r=s zlog._eval_is_finitecC|jddjSNrr"rrr r r!rr2zlog._eval_is_extended_positivecCrr)r.r>rr r r!rDr2zlog._eval_is_zerocCrr)r.Zis_extended_nonnegativerr r r!_eval_is_extended_nonnegativer2z!log._eval_is_extended_nonnegativerc sddlm}m}m}m}m} ddlm|st|}|j d|kr#|S|j d} t dt d} } | | || } | dur_| | | | } } | dkr_| |s_| |s_t| | |} | Sdd}z| |\}}| j|||d}Wn&tttfy| j||d}|jrd 7| j||d}|jsYnw| |\}}|||||d }| tr| |}t||r||||\}}|jst||||||Sfd d }i}t|D]}|||\}}||tj|||<qtj} i}|}| |krAd |  | }|D]}||tj|||||<q|||}| tj7} | |kst|||}|D] }|||||7}qK|dkrg|j d ||}|j!r|j"r||dkr|d |tj#8}||||S)Nr)rdrrrr)productkrc Sstjtj}}t|D]/}||r7|\}}||kr6z||WSty5|tjfYSwq ||9}q ||fSr) rr+rrrhasr/leadtermr)rrPrrrrsr r r! coeff_exps    z$log._eval_nseries..coeff_exprr"csNi}||D]\}}||}|kr$||tj||||||<q|Sr)rrr)Zd1Zd2rie1Ze2exrrr r!muls"zlog._eval_nseries..mulrr{)$rRrdrrrr itertoolsrr%r.rmatchrrrrrr rrrrrrTrrrrrrrr+dirrr*r)rrPrrrrdrrrrr<rrrrrrJr@r_dr!ZptermsrZco1rrZpkrrrir r r!rst     "   zlog._eval_nseriesc Csddlm}m}m}|jd}|j||d}||d} | tj ur6|dur6|j |d||j r2dndd} | tj tj fvrDtd|| dkrP|tj|S|dkrZ|||}| jrq| j rq||j rq|| d |tjS||S) Nr)rrdra)r-+)r$rr"r{)rRrrdrar.Ztogetherrrrrrr*rCr~r r+r$rr,r) rrPrrrrdrarr<Zx0r r r!r s    zlog._eval_as_leading_termrYrrrrj)rZr[r\rmrrr]r^rxr(rrr/rrrr rrrBrrlrr=rrDrrrr r r r!r%Es2     : "   Vr%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 ========== .. 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