a khr@sddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z mZmZmZmZmZddlmZmZmZddlmZdd lmZmZmZmZdd lmZdd l m!Z!dd l"m#Z#dd l$m%Z%m&Z&ddl'm(Z(ddl)m*Z*ddl+m,Z,m-Z-m.Z.m/Z/m0Z0ddl1m2Z2ddl3m4Z4m5Z5ddl6m7Z7Gddde Z8Gddde8Z9GdddeZ:Gddde8e:dZ;ddZed#d$Z?d%S)&) annotations)product)Add)cacheit)Expr)DefinedFunctionArgumentIndexError expand_log expand_mul FunctionClass PoleErrorexpand_multinomialexpand_complex) fuzzy_and fuzzy_notfuzzy_or)Mul)IntegerRationalpiI)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-T/var/www/auris/lib/python3.9/site-packages/sympy/functions/elementary/exponential.pyr*(sz ExpBase.kindcCstS)z= Returns the inverse function of ``exp(x)``. logr,argindexr-r-r.inverse,szExpBase.inversecCsP|js|tjfS|j}|j}|s0| js0|}|rFtj|| 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) )is_commutativerOner) is_negativecould_extract_minus_signfunc)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)Lsz ExpBase.expcCs|dt|jfS)z7 Returns the 2-tuple (base, exponent). r/)r9rr;r+r-r-r. as_base_expSszExpBase.as_base_expcCs||jSr()r9r)Zadjointr+r-r-r. _eval_adjointYszExpBase._eval_adjointcCs||jSr()r9r) conjugater+r-r-r._eval_conjugate\szExpBase._eval_conjugatecCs||jSr()r9r)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,rr-r-r._eval_is_finitebszExpBase._eval_is_finitecCsH|j|j}|j|jkr>|jj}|r(dS|jjrDt|rDdSn|jSdSrA)r9r;r)is_zero is_rationalr)r,szr-r-r._eval_is_rationalls  zExpBase._eval_is_rationalcCs |jtjuSr()r)rNegativeInfinityr+r-r-r. _eval_is_zerowszExpBase._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.rQzs zExpBase._eval_powerc s|ddlm}ddlm}jd}|jrH|jrHtfdd|jDSt ||rr|jrr| |j g|j RS |S)Nr)Product)Sumc3s|]}|VqdSr()r9).0xr+r-r. z1ExpBase._eval_expand_power_exp..) Zsympy.concrete.productsrUZsympy.concrete.summationsrVr;is_Addr5rZfromiter isinstancer9functionlimits)r,hintsrUrVrr-r+r._eval_expand_power_exps    zExpBase._eval_expand_power_expN)r/)__name__ __module__ __qualname__Z unbranchedrComplexInfinity_singularitiespropertyr*r4r:r)r<r=r?r@rGrLrNrQr`r-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}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-r.rks  zexp_polar._eval_evalfcCs||jd|Srh)r9r;)r,rRr-r-r.rQszexp_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|Srh)r;rr6r'r<r+r-r-r.r<s zexp_polar.as_base_expN) rarbrc__doc__is_polar is_comparablerirkrQrqr<r-r-r-r.rgs%rgc@seZdZddZdS)ExpMetacCs&t|jjvrdSt|to$|jtjuS)NT)r) __class____mro__r\rbaserExp1)clsinstancer-r-r.__instancecheck__s zExpMeta.__instancecheck__N)rarbrcr|r-r-r-r.rusrucseZdZdZd+ddZddZeddZed d Z e e d d Z d,ddZ fddZddZddZddZddZd-ddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*ZZS).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 r/cCs|dkr |St||dS)z@ Returns the first derivative of this function. r/N)rr2r-r-r.fdiffsz exp.fdiffcCsddlm}m}|jd}|jrttj}||| fvr>tjS| t t}|r|| d|r|| |rvtj S|||rtjS|| |tjrt S|||tjrtSdS)Nr)askQ)Zsympy.assumptionsr~rr;is_MulrrInfinityNaNas_coefficientrintegerZevenr6Zodd NegativeOneHalf)r,Z assumptionsr~rrZIoocoeffr-r-r. _eval_refines"  zexp._eval_refinecCs~ddlm}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|tt}|rd|j rr|j!r>> 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,deepr_rrr!r r-r-r.rszexp.as_real_imagcs|jrt|jt|j}n|tjur0|jr0t}t|tsD|tjurbdd}t |||||S|tur|js||j ||St ||S)NcSs&|jst|tr"t|ddiS|S)NrPF)is_Powr\r)rr<)rr-r-r.s z exp._eval_subs..) rr)r1rxrryZ is_Functionr\r _eval_subs_subssuper)r,oldnewfrvr-r.rszexp._eval_subscCsB|jdjrdS|jdjr>td t|jdt}|jSdS)NrTr)r;rp is_imaginaryrrrrr,Zarg2r-r-r.rqs   zexp._eval_is_extended_realcCsdd}t||jdS)Ncss|jV|jVdSr() is_complexrCrr-r-r.complex_extended_negativesz7exp._eval_is_complex..complex_extended_negativer)rr;)r,rr-r-r._eval_is_complexszexp._eval_is_complexcCs@|jttjrdSt|jjr<|jjr,dS|jtjr|jjr|jdtjuS|jjr:t |jdt}|jSdSrh) r)rpr;rrMrrrrrr-r-r._eval_is_extended_positives zexp._eval_is_extended_positiverc sddlmddlm}ddlm}ddlm}ddlm }|j } | j |||d} | j r`d| S|| |d} | tjur||||S| tjur|S| jrtd |tfd d | jDr|Std } |} z|| j||d |}Wnttfyd}Yn0|r&|dkr&|||} t | | | }t | || | | }|durd|t|ini}|||kr||S|r|dkr||| | ||||d|7}n||| | ||7}|}||ddd}dd}td|gd}|tj|t tj|}|S)Nrsignceiling)limitOrderpowsimprlogxr/Cannot expand %s around 0c3s|]}t|VqdSr()r\)rWrrr-r.rYrZz$exp._eval_nseries..trTr)rcombinecSs|jo|jdvS)N))rq)rXr-r-r.r rZz#exp._eval_nseries..w)Z properties)!$sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrZsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimprr) _eval_nseriesis_OrderremoveOrrMrrBr anyr;ras_leading_termgetnNotImplementedError_taylorsubsr1rrreplacerr)r,rXrrcdirrrrrrZ arg_seriesarg0rZntermscfZ exp_seriesrrepZ simpleratrr-rr.rsP          (zexp._eval_nseriescCsNg}d}t|D]4}|||jd|}|j||d}||qt|S)Nrr)rangerr;nseriesrrr)r,rXrlgrmr-r-r.rs z exp._taylorcCsddlm}|jdj||d}||d}|tjur@tjSt||rjt |tj krbt | St |S|tjur| |d}|j durt |Std|dS)NrrrFr)Zsympy.calculus.utilrr;cancelrrrrr\r!rr)rrBr )r,rXrrrrrr-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.s zexp._eval_rewrite_as_sincKs0ddlm}|t|t|t|tdS)Nr)rr)rrrr)r,rrrr-r-r._eval_rewrite_as_cos2s zexp._eval_rewrite_as_coscKs,ddlm}d||dd||dS)Nr)tanhr/r)Z%sympy.functions.elementary.hyperbolicr)r,rrrr-r-r._eval_rewrite_as_tanh6s zexp._eval_rewrite_as_tanhcKslddlm}m}|jrh|tt}|rh|jrh|t||t|}}t||sht||sh|t|SdS)Nr)rr) rrrrrrrrr\)r,rrrrrZcosineZsiner-r-r._eval_rewrite_as_sqrt:s zexp._eval_rewrite_as_sqrtcKs<|jr8dd|jD}|r8t|djd||dSdS)NcSs(g|] }t|trt|jdkr|qS)r/)r\r1lenr;)rWrr-r-r. ErZz,exp._eval_rewrite_as_Pow..r)rr;rr)r,rrZlogsr-r-r._eval_rewrite_as_PowCszexp._eval_rewrite_as_Pow)r/)T)r)rarbrcrrr}r classmethodrrfrx staticmethodrrrrrqrrrrrrrrrrr __classcell__r-r-rr.r)s0  i     /  r)) metaclasscCsR|jtdd\}}|dkr(|jr(||fS|t}|rJ|jrJ|jrJ||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_independentris_realr)exprr_i_r-r-r.match_real_imagJs  rc@seZdZUdZded<ejejfZd*ddZ d+ddZ e d,d d Z e ed d Zd-ddZddZd.ddZddZddZddZddZddZdd Zd!d"Zd#d$Zd/d&d'Zd(d)Zd S)0r1a The natural logarithm function `\ln(x)` or `\log(x)`. Explanation =========== Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. ``log`` represents the principal branch of the natural logarithm. As such it has a branch cut along the negative real axis and returns values having a complex argument in `(-\pi, \pi]`. Examples ======== >>> from sympy import log, sqrt, S, I >>> log(8, 2) 3 >>> log(S(8)/3, 2) -log(3)/log(2) + 3 >>> log(-1 + I*sqrt(3)) log(2) + 2*I*pi/3 See Also ======== exp z tuple[Expr]r;r/cCs$|dkrd|jdSt||dS)z? Returns the first derivative of the function. r/rN)r;rr2r-r-r.r}sz log.fdiffcCstS)zC Returns `e^x`, the inverse function of `\log(x)`. )r)r2r-r-r.r4sz log.inverseNcCsddlm}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|jSt|tr|jjr|jSt|tr|jjrt|j\}}|rf|jrf|dt;}|tkr|dt8}|t|tddSn|t|t rt!|jSt||rP|j"j#r,|t |j"t |j$S|j"j rH|tjt |j$StjSnt||rf|%|S|jr|j&rtt|| S|tjurtjS|tj urtjS|j rtjS|j's:|(t}|dur:|tjurtjS|tjurtjS|jr:|j)r tttj*||St ttj*|| S|jr|j+r|j,tdd\}} |j&rt|d 9}| d 9} t| dd} | j,td d\}}|(t}|j-r|r|j-r|j-r|j r |j#rtttj*|||S|j&rt ttj*||| Sndd l.m/} ||0} | 0} t1} | | vr| |t2| }|j#rj||t| | S||t| | tSnP| | vr| |t2| }|j#r||t| |  S||tt| | SdS) Nrrrr/rFrrrT)ratsimp)3rrrrrrrrdr$r1 ValueErrorryrrHr6rrrMrrrrrxr)rpr\rrrtrr rrgrrrrrr7r[ris_nonnegativerrrrZsympy.simplifyrr_log_atan_tabler")rzrrxrrrrrrZarg_rrt1Z atan_tablemodulusr-r-r.rs                                   zlog.evalcGsddlm}|dkrtjSt|}|dkr.|S|rb|d}|durb|| |||ddddSdd |d ||d|dS) zV Returns the next term in the Taylor series expansion of `\log(1+x)`. rrrNr/Tr)rr)rrrrr)rrXrrrr-r-r.rs  zlog.taylor_termTcKsxddlm}m}|dd}|dd}t|jdkrLt|j|j||dS|jd}|jrt |}d} d} |dur|\}} ||} |rt |}|| vrt d d | D} | dur| | Sn|jrt|jt|jS|jrg} g} |jD]} |s| js| jrR|| }t|trF| || jfi|n | |q| jr~|| }| || tjq| | qt| tt| S|jst|tr:|s|jjr|j js|jdjr|jdj!s|j jrn|j }|j}||}t|tr,t"||jfi|St"||Sn4t||rn|sV|j#jrn|t|j#g|j$RS||S) Nr)rVrUforceFfactorr)rr r/css|]\}}|t|VqdSr(r0)rWvalrr-r-r.rY7rZz'log._eval_expand_log..)%Zsympy.concreterVrUgetrr;r r9Z is_Integerr%r&keyssumitemsrr1rrrrrsr\r_eval_expand_logr7rrrrrr)rprxis_nonpositiverr]r^)r,rr_rVrUr rrrZlogargrrZnonposrXrrSrTr-r-r.r$sh             (    zlog._eval_expand_logcKsddlm}m}m}t|jdkr:||j|jfi|S|||jdfi|}|drf||}||dd}t||g|ddS) Nr)r simplifyinversecombinerr4TrZmeasure)key)rr rrrr;r9r)r,rr rrrr-r-r._eval_simplify]s zlog._eval_simplifycKs|jd}|r&|jdj|fi|}t|}||kr@|tjfSt|}|ddrvd|d<t|j|fi||fSt||fSdS)a Returns this function as a complex coordinate. Examples ======== >>> from sympy import I, log >>> from sympy.abc import x >>> log(x).as_real_imag() (log(Abs(x)), arg(x)) >>> log(I).as_real_imag() (0, pi/2) >>> log(1 + I).as_real_imag() (log(sqrt(2)), pi/4) >>> log(I*x).as_real_imag() (log(Abs(x)), arg(I*x)) rr1FcomplexN)r;rr"rrrrr1)r,rr_ZsargZsarg_absZsarg_argr-r-r.rhs   zlog.as_real_imagcCs\|j|j}|j|jkrR|jddjr,dS|jdjrXt|jddjrXdSn|jSdSNrr/TF)r9r;rHrIrr,rJr-r-r.rLs   zlog._eval_is_rationalcCs\|j|j}|j|jkrR|jddjr,dSt|jddjrX|jdjrXdSn|jSdSr)r9r;rHrrrr-r-r.rs   zlog._eval_is_algebraiccCs |jdjSrhr;rDr+r-r-r.rqszlog._eval_is_extended_realcCs|jd}t|jt|jgSrh)r;rrrrH)r,rKr-r-r.rs zlog._eval_is_complexcCs|jd}|jrdS|jSNrF)r;rHrErFr-r-r.rGs zlog._eval_is_finitecCs|jddjSNrr/rr+r-r-r.rszlog._eval_is_extended_positivecCs|jddjSr)r;rHr+r-r-r.rNszlog._eval_is_zerocCs|jddjSr)r;Zis_extended_nonnegativer+r-r-r._eval_is_extended_nonnegativesz!log._eval_is_extended_nonnegativerc$ sddlm}ddlm}ddlm}|jd|krF|durBt|S|S|jd}|ddd} |dkrhd}|||| } t d t d } } | | | | } | dur| | | | } } | dkr| | s| | s|dur| t|n| |} | t| | t|7} | Sd d }z| j | |dd \}}Wnt ttfy| j| |dd}|jrd7| j| |dd}qZz|j | dd\}}Wn.t y|j| ddtj}}Yn0Yn0| || |dj| |dd}| tr ||}t||r|||| \}}|dur>t|n|}|jst||t|||}|}dddddddddd }|jfi|}|s|r|| t| jfi|}n||t|jfi|}||kr|S||||Sfdd}i}t|D]*}||| \}}||tj|||<q"tj } i}|}| |krtj!|  | } |D]$}!||!tj| ||!||!<q~|||}| tj 7} q\t||t|||}|D]}!|||!| |!7}q|j"rt#| dkrddl$m%}"t&| '| D]"\}#}|j(rF|#dkr,qPq,|#dkr|)| \} }|dt*t+|"t#|  d7}|| ||}||||S)Nrrr)rrTZpositiver/krc Sstjtj}}t|D]^}||rn|\}}||krvz||WStyj|tjfYS0q||9}q||fSr() rr6rrrhasr<leadtermr)rrXrr)rrxr-r-r. coeff_exps    z$log._eval_nseries..coeff_exprr)rrr)rF) rr1mulZ power_expZ power_baseZ multinomialbasicr rcsNi}t||D]:\}}||}|kr||tj||||||<q|Sr()rrrr)d1Zd2roe1e2exrr-r.r's $zlog._eval_nseries..mul Heaviside),rrrrsympy.core.symbolrr;r1rrmatchr#r$rrr rrrrrrrr)r\rrrr8rrrr6rr7r 'sympy.functions.special.delta_functionsr. enumeratelseriesras_coeff_exponentrr)$r,rXrrrrrrrrrKr"rrr%rrSrJr_droZ_resZlogflagsrr'ZptermsrZco1r*rpkrr,r.rmr-rr.rs      "&&   "  "     zlog._eval_nseriescCs|jd}tddd}|dkr&d}||||}z|j||dd\}}Wn*tyx|j|||d} t| YS0||r||||}|dkrt d|t|S|t j kr|t j kr|t j j||dSt||t|} |durt|n|}| ||7} |j rt|dkrdd lm} t||D]"\} } | jrX| d kr>qbq>| d kr| |\}}| d tt| t| d7} | S) NrrTr!r/r&rrr-r/r0)r;Ztogetherrrr$rrr1r#r rr6rr7r r3r.r4r5rr6rr)r,rXrrrrrKcrTrror.rmrrr7r-r-r.r's:         zlog._eval_as_leading_term)r/)r/)N)T)T)r)rarbrcrr__annotations__rrrdrer}r4rrrrrrrrrLrrqrrGrrNr rrr-r-r-r.r1_s. !   }  9    tr1cszeZdZdZeejddd ejfZe dddZ dd d Z d d Z d dZ ddZddZdfdd 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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