OtjrddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z mZmZmZmZmZddlmZmZmZddlmZdd lmZmZmZmZdd lmZdd l m!Z!dd l"m#Z#dd l$m%Z%m&Z&ddl'm(Z(ddl)m*Z*ddl+m,Z,m-Z-m.Z.m/Z/m0Z0ddl1m2Z2ddl3m4Z4m5Z5ddl6m7Z7Gdde Z8Gdde8Z9GddeZ:Gdde8e:Z;dZ<Gdde Z=Gd d!e Z>ed"Z?d#S)$) 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) factorintceZdZdZejfZedZddZ dZ edZ dZ dZ d Zd Zd Zd Zd ZdZdZdS)ExpBaseTc|jjSN)expkindselfs d/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/functions/elementary/exponential.pyr-z ExpBase.kind(s x}ctS)z= Returns the inverse function of ``exp(x)``. logr/argindexs r0inversezExpBase.inverse,  r1c|js|tjfS|j}|j}|s| js|}|r"tj|| 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,neg_exps r0as_numer_denomzExpBase.as_numer_denom2s{ " ; h/ 51 52244G  *5$))SD//) )QU{r1c|jdS)z7 Returns the exponent of the function. r)argsr.s r0r,z ExpBase.expLs y|r1cH|dt|jfS)z7 Returns the 2-tuple (base, exponent). r2)r?rrCr.s r0 as_base_expzExpBase.as_base_expSsyy||S$)_,,r1cZ||jSr+)r?r,adjointr.s r0 _eval_adjointzExpBase._eval_adjointYs"yy))++,,,r1cZ||jSr+)r?r, conjugater.s r0_eval_conjugatezExpBase._eval_conjugate\"yy++--...r1cZ||jSr+)r?r, transposer.s r0_eval_transposezExpBase._eval_transpose_rLr1cX|j}|jr|jrdS|jrdS|jrdSdSNTF)r, is_infiniteis_extended_negativeis_extended_positive is_finiter/rs r0_eval_is_finitezExpBase._eval_is_finitebsLh ? ' t' u = 4  r1c|j|j}|j|jkr1|jj}|rdS|jjrt |rdSdSdS|jSrQ)r?rCr,is_zero is_rationalr)r/szs r0_eval_is_rationalzExpBase._eval_is_rationallst DIty ! 6TY   A t" y|| u    = r1c(|jtjuSr+)r,rNegativeInfinityr.s r0 _eval_is_zerozExpBase._eval_is_zerowsx1---r1cz|\}}tjt||d|S)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)rEr _eval_power)r/otherbes r0rdzExpBase._eval_powerzs:!!1s1a%888%@@@r1c @ddlm}ddlm}jd}|jr,|jr%tjfd|jDSt||r-|jr&| |j g|j RS |S)Nr)Product)Sumc3BK|]}|VdSr+)r?).0xr/s r0 z1ExpBase._eval_expand_power_exp..s-?? ! ??????r1) sympy.concrete.productsrisympy.concrete.summationsrjrCis_Addr;rfromiter isinstancer?functionlimits)r/hintsrirjrs` r0_eval_expand_power_expzExpBase._eval_expand_power_exps333333111111il : A#, A<????ch????? ? S ! ! Ac&8 A7499S\22@SZ@@@ @yy~~r1Nr2)__name__ __module__ __qualname__ unbranchedrComplexInfinity_singularitiespropertyr-r8rAr,rErHrKrOrWr]r`rdrwr1r0r)r)#sJ')N X 4X --- ---////// ! ! !...AAA r1r)c8eZdZdZdZdZdZdZdZdZ dZ d S) 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 TFcPtt|jdSNr)r,r"rCr.s r0 _eval_Abszexp_polar._eval_Abss2dil##$$$r1cBt|jd} |t kp |tk}n#t$rd}YnwxYw|r|St |jd|}|dkr"t|dkrt |S|S)z. Careful! any evalf of polar numbers is flaky rT)r!rCr TypeErrorr, _eval_evalfr")r/precibadress r0rzexp_polar._eval_evalfs ty|   8%q2vCC   CCC   K$)A,++D11 q55RWWq[[c77N s4 AAcH||jd|zSr)r?rC)r/res r0rdzexp_polar._eval_powersyy1e+,,,r1c.|jdjrdSdS)NrT)rCis_extended_realr.s r0_eval_is_extended_realz exp_polar._eval_is_extended_reals" 9Q< ( 4  r1ct|jddkr|tjfSt|Sr)rCrr<r)rEr.s r0rEzexp_polar.as_base_exps3 9Q<1  ; ""4(((r1N) ryrzr{__doc__is_polar is_comparablerrrdrrErr1r0rrsv##JHM%%%   ---)))))r1rceZdZdZdS)ExpMetac|t|jjvrdSt|to|jt juS)NT)r, __class____mro__rsrbaserExp1)clsinstances r0__instancecheck__zExpMeta.__instancecheck__s6 ($, , ,4(C((DX]af-DDr1N)ryrzr{rrr1r0rrs(EEEEEr1rceZdZdZddZdZedZedZ e e dZ dd Z fd Zd Zd Zd ZdZddZdZdZdZdZdZdZdZxZS)r,a9 The exponential function, :math:`e^x`. Examples ======== >>> from sympy import exp, I, pi >>> from sympy.abc import x >>> exp(x) exp(x) >>> exp(x).diff(x) exp(x) >>> exp(I*pi) -1 Parameters ========== arg : Expr See Also ======== log r2c2|dkr|St||)z@ Returns the first derivative of this function. r2)r r6s r0fdiffz exp.fdiffs" q==K$T844 4r1cddlm}m}|jd}|jr t t jz}||| fvr t jS|j tt z}|r|| d|zr|| |r t j S|||r t jS|| |t jzrt S|||t jzr t SdSdSdSdS)Nr)askQ)sympy.assumptionsrrrCis_MulrrInfinityNaNas_coefficientrintegerevenr<odd NegativeOneHalf)r/ assumptionsrrrIoocoeffs r0 _eval_refinezexp._eval_refinesU,,,,,,,,il : !AJ,CsSDk!!u &C&r!t,,E !3qyy5))**!s166%==))! u QUU5\\**! },QVVEAFN3344! !r QUU516>2233!  ! ! ! !!!!!r1cddlm}ddlm}ddlm}ddlm}t||r|j Stj rttj|S|jr}|tjur tjS|jr tjS|tjur tjS|tjur tjS|tjur tjSn|tjur tjSt|t.r |jdSt||r0|t|jt|jSt||r|j|S|jr |jt<t>z}|rd|zj r^|j!r tjS|j"r tj#S|tj$zj!rt> S|tj$zj"rt>Sn8|j%r1|dz}|dkr|dz}||kr||t<zt>zS|j&\}}|tjtjfvr|j'r|tjur| }tQ|jr|tjur tjStQ|j)r'tU|tjur tjStQ|j+r tjSdS|gd} } tYj-|D]T} || } t| t.r| | jd} 2dS| j.r| /| RdS| r | tY| zndS|j0rg} g}d}|jD]}|tjur|/|&||}t||rJ|jd|kr#|/|jdd }u|/|| /|| s|rtY| |tc|d zS|jr tjSdS) Nr AccumBounds) MatrixBaseSetExpr logcombinerr2FTrb)2sympy.calculusrsympy.matrices.matrixbasersympy.sets.setexprrsympy.simplify.simplifyrrsr,r exp_is_powrrr is_NumberrrYr<rr_Zeror}r5rCminmax _eval_funcrrrr is_integeris_evenis_oddrr is_Rational as_coeff_Mul is_numberr" is_positiver!r=r make_argsrappendrqr)rrrrrrrncoefftermscoeffslog_termtermterm_outadd argchangedanewas r0evalzexp.evals......888888......666666 c: & &_ @3799   )] @qvs## # ][ @ae||u  u v  ""z!***v + A% % %5L S ! !N @8A;  [ ) )L @;s37||S\\:: : W % %J @!3>#&& & ZH @&C&r!t,,E 0eG'0}! u ! },!&.1! !r !&.0! !&0"QYFzz! "s6"9Q;///,3+--LE5+QZ888?& 222!&%yy(%U!&-@-@ u %yy,1E!&1H1H 00%yy,& v t %wHF e,, " 4((eS)) '#(:a=#tt' MM$''''44-5?8S&\))4 ? Z @CCJX % %::JJqMMMs1vvdC((%y|q(( 49Q<000%)  1 JJt$$$$ @j @CyS#Y!?!?!??? ; 5L  r1ctjS)z? Returns the base of the exponential function. )rrr.s r0rzexp.base}s v r1c|dkr tjS|dkr tjSt|}|r|d}|||z|z S||zt |z S)zJ Calculates the next term in the Taylor series expansion. r)rrr<rr)nrmprevious_termsps r0 taylor_termzexp.taylor_termsi q556M 665L AJJ  !r"A}1uqy !tIaLL  r1Tc ddlm}m}|jd\}}|r|j|fi|}|j|fi|}||||}}t ||zt ||zfS)aJ Returns this function as a 2-tuple representing a complex number. Examples ======== >>> from sympy import exp, I >>> from sympy.abc import x >>> exp(x).as_real_imag() (exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x))) >>> exp(1).as_real_imag() (E, 0) >>> exp(I).as_real_imag() (cos(1), sin(1)) >>> exp(1+I).as_real_imag() (E*cos(1), E*sin(1)) See Also ======== sympy.functions.elementary.complexes.re sympy.functions.elementary.complexes.im r)cossin)(sympy.functions.elementary.trigonometricrrrC as_real_imagexpandr,)r/deeprvrrr"r!s r0rzexp.as_real_imags0 FEEEEEEE1**,,B  *4))5))B4))5))B3r77CCGGSB SWWS[))r1c|jr*t|jt|jz}n|tjur|jrt}t|ts|tjur+d}tj |||||S|tur%|js||j ||zSt ||S)Ncz|jst|trt|ddin|S)NrcF)is_Powrsr,rrE)rs r0z exp._eval_subs..s@7&q#..7#q}}????56r1) rr,r5rrr is_Functionrsr _eval_subs_subssuper)r/oldnewfrs r0rzexp._eval_subss : cgc#(mm+,,CC AF]]s]C c3   83!&==77A>!!D''11S66377 7 #::co:sC000 0ww!!#s+++r1c|jdjrdS|jdjr5td tz|jdzt z }|jSdS)NrTr)rCr is_imaginaryrrrrr/arg2s r0rzexp._eval_is_extended_reals\ 9Q< ( 4 Yq\ & aDD519ty|+b0D<   r1cNd}t||jdS)Nc3.K|jV|jVdSr+) is_complexrS)rs r0complex_extended_negativez7exp._eval_is_complex..complex_extended_negatives). * * * * * *r1r)rrC)r/rs r0_eval_is_complexzexp._eval_is_complexs3 + + +11$)A,??@@@r1c|jtz tz jrdSt |jjr$|jjrdS|jtz jrdSdSdSrQ)r,rrrZrrY is_algebraicr.s r0_eval_is_algebraiczexp._eval_is_algebraicsn HrMA  * 4 TX% & & x$ u(R-, u     r1c|jjr|jdtjuS|jjr%t |jdztz }|jSdSr) r,rrCrr_rrrrrs r0_eval_is_extended_positivezexp._eval_is_extended_positivesY 8 $ 9Q.s-::z#t$$::::::r1tr Tr,rcombinec"|jo|jdvS)N))rq)rms r0rz#exp._eval_nseries.. sam@y0@r1w) properties)!$sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimpr r, _eval_nseriesis_OrderremoveOrr_rrRr anyrCras_leading_termgetnNotImplementedError_taylorsubsr5rrreplacerr)r/rmrr cdirrrrr r arg_seriesarg0rntermscf exp_seriesrrep simpleratrrs @r0rzexp._eval_nseriess >=====??????------,,,,,,222222h&S&qAD999   "z> !uZ''))1a00 1% % %5Aq>> ! 1:  K   B74@AA A :::: ::: : : K #JJ *s*14888!<<AACCBB#Y/   BBB   #"q&&WQrT]]FVV^^Av.. IIjooad):;; ; $ 0tSVVnnb 66#;;$  H  2"q&&  T)A-q11!r!tQh-? ?AA  T)A-q11 1A HHJJ GAD% 0 0 0@@ ) - - - IIamQ&q}a7G(H(H I Is.DD$#D$cg}d}t|D]b}|||jd|}|||}||ct |S)Nr)r)rangerrCnseriesrr!r)r/rmrlgrs r0r&z exp._taylorsy  q " "A  DIaL!44A !q !!A HHQYY[[ ! ! ! !Awr1cddlm}|jd||}|j|d}|t jur t jSt||r>>>>>DDQKK!dd3q5kk/22r1c &ddlm}m}|jry|jt t z}|r\|jrW|t |z|t |z}}t||s#t||s|t |zzSdSdSdSdSdS)Nr)rr) rrrrrrrrrs)r/rr<rrrcosinesines r0_eval_rewrite_as_sqrtzexp._eval_rewrite_as_sqrt:sEEEEEEEE : +CIbdOOE + +"s2e8}}cc"U(mm!&#..+z47M7M+!AdF?*  + + + + + +++++r1c |jrHd|jD}|r7t|djd|j|dSdSdS)Nclg|]1}t|tt|jdk/|2Srx)rsr5lenrC)rlrs r0 z,exp._eval_rewrite_as_Pow..Es:SSS!:a+=+=S#af++QRBRBRABRBRBRr1r)rrCrr)r/rr<logss r0_eval_rewrite_as_Powzexp._eval_rewrite_as_PowCsm : @SSsxSSSD @47<?ICId1g,>,>??? @ @ @ @r1rxTr)ryrzr{rrr classmethodrrr staticmethodrrrrrrrrrr&r:r=r?rCrGrM __classcell__rs@r0r,r,s45555!!!(gg[gRX   ! ! W\ !****@ , , , , ,   AAA    ----^>>>*000000333+++@@@@@@@r1r,) metaclassc|td\}}|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. Tas_Addr)NN)as_independentris_realr)exprr_i_s r0match_real_imagr]Jss 4 0 0FB Qww2:wBx  1  B bjRZBx|r1ceZdZUdZded<ejejfZddZ ddZ e ddZ e ed Zdd Zd Zdd ZdZdZdZdZdZdZdZdZddZdZdS)r5a 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]rCr2cN|dkrd|jdz St||)z? Returns the first derivative of the function. r2r)rCr r6s r0rz log.fdiffs- q==TYq\> !$T844 4r1ctS)zC Returns `e^x`, the inverse function of `\log(x)`. )r,r6s r0r8z log.inverser9r1Nc ddlm}ddlm}t |}|t |}|dkr|dkr t jSt jS t||}|r(|t|||zz t|z zSt|t|z S#t$rYnwxYw|t j ur||||z S||S|j r|j r t jS|t jur t jS|t jur t jS|t jur t jS|t jur t jS|jr|jdkr||j S|jr&|jt j ur|jjr|jSt3|t.r|jjr|jSt3|t.rn|jjrbt7|j\}}|rH|jrA|dt:zz}|t:kr |dt:zz}|t=|t>zdzSnt3|t@rtC|jSt3||r||j"j#r0|t|j"t|j$S|j"j r(|t jt|j$St jSt3||r|j%|S|jrW|j&rt:t>z|| zS|t jur t jS|t j ur t jS|j r t jS|j's|j(t>}||t jur t jS|t jur t jS|jrY|j)r(t:t>zt j*z||zSt: t>zt j*z|| zS|jr|j+r|j,t>d\}} |j&r |d z}| d z} t=| d} | ,t>d \}}|(t>}|j-r|r|j-r|j-r|j rh|j#r+t:t>zt j*z|||zzS|j&r-t: t>zt j*z||| zzSdSdd l.m/} ||z 0} | 0} tc} | | vrb| |te| z}|j#r||t>| | zzS||t>| | t:z zzS| | vro| |te| z}|j#r||t>| |  zzS||t>t:| | z zzSdSdSdSdSdSdSdS) Nrrrr2rFrrVrT)ratsimp)3rrrrrrrr}r%r5 ValueErrorrrrYr<rrr_rrrrrr,rrsrr]rrr rrr rrrrr=rqris_nonnegativerrrXrYsympy.simplifyrcr9_log_atan_tabler#)rrrrrrr[r\rarg_rcrt1 atan_tablemoduluss r0rzlog.evals............cll  4==Dqyy!885L,, !s++.s3q=11CII===s88CII--    16!!s3xxD ))s3xx = #{ #((v  ""z!***z!u  #SUaZZCE {" : #(af,,1I,7N c3   'CG$< '7N S ! ! 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