MZdƦddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z mZmZmZmZmZmZddlmZmZmZddlmZdd lmZmZmZmZmZdd l m!Z!dd l"m#Z#dd l$m%Z%dd l&m'Z'm(Z(ddl)m*Z*ddl+m,Z,ddl-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4ddl5m6Z6m7Z7ddl8m9Z9Gdde Z:Gdde:Z;GddeZ<Gdde:e<Z=dZ>Gdde Z?Gd d!e Z@ed"ZAy#)$)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) factorintceZdZdZej fZedZddZ dZ edZ dZ dZ dZd Zd Zd Zd Zd ZdZy)ExpBaseTc.|jjSN)expkindselfs H/usr/lib/python3/dist-packages/sympy/functions/elementary/exponential.pyr.z ExpBase.kind(sxx}}ctS)z= Returns the inverse function of ``exp(x)``. logr0argindexs r1inversezExpBase.inverse,  r2c|j}|j}|s| js|j}|r"tj|j | 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)r0r-neg_exps r1as_numer_denomzExpBase.as_numer_denom2s[ hh//11224G 55$))SD/) )QUU{r2c |jdS)z7 Returns the exponent of the function. r)argsr/s r1r-z ExpBase.expJs yy|r2cH|jdt|jfS)z7 Returns the 2-tuple (base, exponent). )r>rrBr/s r1 as_base_expzExpBase.as_base_expQsyy|S$))_,,r2cT|j|jjSr,)r>r-adjointr/s r1 _eval_adjointzExpBase._eval_adjointWsyy))+,,r2cT|j|jjSr,)r>r- conjugater/s r1_eval_conjugatezExpBase._eval_conjugateZyy++-..r2cT|j|jjSr,)r>r- transposer/s r1_eval_transposezExpBase._eval_transpose]rLr2c|j}|jr|jry|jry|jryyNTF)r- is_infiniteis_extended_negativeis_extended_positive is_finiter0r s r1_eval_is_finitezExpBase._eval_is_finite`s9hh ??'''' == r2c|j|j}|j|jk(r=|jj}|ry|jjr t |ryyy|jSrQ)r>rBr-is_zero is_rationalr)r0szs r1_eval_is_rationalzExpBase._eval_is_rationaljsc DIItyy ! 66TYY  A""y|(4"== r2c:|jtjuSr,)r-rNegativeInfinityr/s r1 _eval_is_zerozExpBase._eval_is_zerousxx1----r2cl|j\}}tjt||d|S)z;exp(arg)**e -> exp(arg*e) if assumptions allow it. Fevaluate)rEr _eval_power)r0otherbes r1rdzExpBase._eval_powerxs0!1s1a%8%@@r2c dddlm}ddlm}jd}|j r4|j r(tjfd|jDSt||r8|j r,|j|jg|jSj|S)Nr)Product)Sumc3@K|]}j|ywr,)r>).0xr0s r1 z1ExpBase._eval_expand_power_exp..s? ! ?s) sympy.concrete.productsrisympy.concrete.summationsrjrBis_Addis_commutativerfromiter isinstancer>functionlimits)r0hintsrirjr s` r1_eval_expand_power_expzExpBase._eval_expand_power_exp~s31iil ::#,,<<?chh?? ? S !c&8&8499S\\2@SZZ@ @yy~r2NrD)__name__ __module__ __qualname__ unbranchedrComplexInfinity_singularitiespropertyr.r8r@r-rErHrKrOrWr]r`rdrxr2r1r*r*#ssJ'')N  0 - -// !.A r2r*c6eZdZdZdZdZdZdZdZdZ dZ y ) 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 TFcDtt|jdSNr)r-r#rBr/s r1 _eval_Abszexp_polar._eval_Abss2diil#$$r2ct|jd} |t kxs |tkD}|r|St |jdj |}|dkDrt|dkr t |S|S#t$rd}YXwxYw)z. Careful! any evalf of polar numbers is flaky rT)r"rBr TypeErrorr- _eval_evalfr#)r0precibadress r1rzexp_polar._eval_evalfs tyy|  8%q2vC K$))A,++D1 q5RWq[c7N  C sA:: BBcD|j|jd|zSr)r>rB)r0res r1rdzexp_polar._eval_powersyy1e+,,r2c8|jdjryy)NrT)rBis_extended_realr/s r1_eval_is_extended_realz exp_polar._eval_is_extended_reals 99Q< ( ( )r2ct|jddk(r|tjfStj |Sr)rBrr=r*rEr/s r1rEzexp_polar.as_base_exps1 99Q<1 ; ""4((r2N) rzr{r|__doc__is_polar is_comparablerrrdrrErr2r1rrs-#JHM% -)r2rceZdZdZy)ExpMetact|jjvryt|txr|j t juS)NT)r- __class____mro__rtrbaserExp1)clsinstances r1__instancecheck__zExpMeta.__instancecheck__s8 ($$,, ,(C(DX]]aff-DDr2N)rzr{r|rrr2r1rrsEr2rceZdZdZddZdZedZedZ e e dZ ddZ dZd Zd Zd Zd Zdd ZdZddZdZdZdZdZdZy)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 c(|dk(r|St||)z@ Returns the first derivative of this function. rD)r r6s r1fdiffz exp.fdiffs q=K$T84 4r2cvddlm}m}|jd}|jrt t jz}||| fvrt jS|jtt z}|r||jd|zr||j|rt jS||j|rt jS||j|t j zrt S||j|t j zrt Syyyy)Nr)askQ)sympy.assumptionsrrrBis_MulrrInfinityNaNas_coefficientrintegerevenr=odd NegativeOneHalf)r0 assumptionsrrr Ioocoeffs r1 _eval_refinezexp._eval_refines,iil ::AJJ,CsSDk!uu &C&&r!t,Eqyy5)*166%=) uu QUU5\* }},QVVEAFFN34 !r QUU5166>23 4+ r2c ddlm}ddlm}ddlm}ddlm}t||r|jStjrttj|S|jr|tj urtj S|j"rtj$S|tj$urtjS|tj&urtj&S|tj(urtj*S|tj,urtj St|t.r|j0dSt||r/|t|j2t|j4St||r|j6|S|j8rP|j:t<t>z}|rd|zj@r|jBrtj$S|jDrtjFS|tjHzjBrt> S|tjHzjDrS|jJr*|dz}|dkDr|dz}||k7r||t<zt>zS|jL\}}|tj(tj&fvr|jNr|tj(ur| }tQ|j"r"|tj*urtj StQ|jRr+tU|tj*urtj,StQ|jVrtj*Sy|gd} } tYjZ|D]M} || } t| t.r| | j0d} -y| j\r| j_| My| r | tY| zSdS|j`rg} g}d}|j0D]}|tj$ur|j_|'||}t||rE|j0d|k7r!|j_|j0dd }n|j_|| j_|| s|rtY| |tc|d zS|j"rtj$Sy) Nr AccumBounds) MatrixBaseSetExpr logcombinerrDFTrb)2sympy.calculusrsympy.matrices.matricesrsympy.sets.setexprrsympy.simplify.simplifyrrtr-r exp_is_powrrr is_NumberrrYr=rr_Zeror~r5rBminmax _eval_funcrrrr is_integeris_evenis_oddrr is_Rational as_coeff_Mul is_numberr# is_positiver"r;r make_argsrappendrqr)rr rrrrrncoefftermscoeffslog_termtermterm_outadd argchangedanewas r1evalzexp.evals.6.6 c: &3779   ) )qvvs# # ]]aee|uu uu vv  "zz!***vv A%% %55L S !88A;  [ )s377|S\: : W %!3>>#& & ZZ&C&&r!t,EeG''}} uu  }},!&&.11 !r !&&.00 &&"QYFz! "6"9Q;//,3++-LE5++QZZ88?? 2 22!&%y((U!&&-@ uu %y,,E!&&1H 000%y,, vv  %wHF e, "4(eS)'#(::a=#''MM$' .68S&\) ?4 ? ZZCCJXX %:JJqM1vdC(yy|q( 499Q<0%)  1 JJt$ %jCyS#Y!??? ;;55L r2c"tjS)z? Returns the base of the exponential function. )rrr/s r1rzexp.base{s vv r2c|dkrtjS|dk(rtjSt|}|r|d}|||z|z S||zt |z S)zJ Calculates the next term in the Taylor series expansion. r)rrr=rr)nrmprevious_termsps r1 taylor_termzexp.taylor_terms` q566M 655L AJ r"A}1uqy !tIaL  r2c ddlm}m}|jdj \}}|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.trigonometricrrrB as_real_imagexpandr-)r0deeprwrrr#r"s r1rzexp.as_real_imags{0 F1**,B 4)5)B4)5)Br7CGSB SWS[))r2c|jr,t|jt|jz}n$|tj ur|j rt}t|ts|tj ur&d}tj|||||S|tur+|j s||jj||zStj|||S)Ncp|jst|trt|j ddiS|S)NrcF)is_Powrtr-rrE)rs r1z exp._eval_subs..s1Jq#.q}}??56r2) rr-r5rrr is_Functionrtr _eval_subs_subsr)r0oldnewfs r1rzexp._eval_subss ::cggc#((m+,C AFF]sC c3 3!&&=7A>>!D'1S637 7 #:coosC00 0""4c22r2c|jdjry|jdjr6td tz|jdzt z }|j Sy)NrTr)rBr is_imaginaryrrrrr0arg2s r1rzexp._eval_is_extended_realsW 99Q< ( ( YYq\ & &aD519tyy|+b0D<< 'r2cDd}t||jdS)Nc3DK|j|jywr,) is_complexrS)r s r1complex_extended_negativez7exp._eval_is_complex..complex_extended_negatives.. ** *s r)rrB)r0rs r1_eval_is_complexzexp._eval_is_complexs" +1$))A,?@@r2c|jtz tz jryt |jj r6|jj ry|jtz jryyyrQ)r-rrrZrrY is_algebraicr/s r1_eval_is_algebraiczexp._eval_is_algebraics[ HHrMA  * * TXX%% &xx$$((R-,,- 'r2c|jjr|jdtjuS|jj r*t |jdztz }|jSyr) r-rrBrr_rrrrrs r1_eval_is_extended_positivezexp._eval_is_extended_positives] 88 $ $99Q.sK#z#m45Kstr Tr-rcombinec:|jxr|jdvS)N))rq)rms r1rz#exp._eval_nseries..samm@y0@r2w) properties) $sympy.functions.elementary.complexesr#sympy.functions.elementary.integersrsympy.series.limitsrsympy.series.orderrsympy.simplify.powsimpr r- _eval_nseriesis_OrderremoveOrr_ranyrBras_leading_termgetnNotImplementedErrorr _taylorsubsr5rrreplacerr)r0rmrr cdirrrrr r arg_seriesarg0rntermscf exp_seriesrrep simpleratrrs @r1rzexp._eval_nseriess >?-,2hh&S&&qAD9   z> !Z'')1a0 1%% %Aq> ! 1:: K KK KK #J *s**148!<AACB "q&QrT]FV^^Av. Ijooad):; ; $ 0tSVnb 66#;$ H "q&  T)A-q1!r!tQh-? ?A  T)A-q1 1A HHJ AD% 0@ ) - IIammQ&q}}a7G(H I'$Y/ B s )H H H cg}d}t|D]T}|j||jd|}|j||}|j |j Vt |S)Nr)r)rangerrBnseriesrr r)r0rmrlgrs r1r%z exp._taylor sj  q "A  DIIaL!4A !q !A HHQYY[ ! "Awr2Ncddlm}|jdjj ||}|j |d}|t jurt jSt||r3t|t jkr t| St|S|t jur|j|d}|jdur t|Std|z)NrrrFCannot expand %s around 0)sympy.calculus.utilrrBcancelr"r&rrrtr#rr-rrRr )r0rmr r(rr r*s r1_eval_as_leading_termzexp._eval_as_leading_terms3iil!!#33AD3Asxx1~ !%%<55L dK ( $x!&& D5z!t9  155=399Q?D   u $t9 3t<==r2c nddlm}|t|ztdz zt|t|zzz S)Nr)rr)rrrr)r0r kwargsrs r1_eval_rewrite_as_sinzexp._eval_rewrite_as_sin*s-@1S52a4< 1S3Z<//r2c nddlm}|t|zt|t|ztdz zzzS)Nr)rr)rrrr)r0r r<rs r1_eval_rewrite_as_coszexp._eval_rewrite_as_cos.s.@1S5zAc!C%"Q$,////r2c Hddlm}d||dz zd||dz z z S)Nr)tanhrDr)%sympy.functions.elementary.hyperbolicrA)r0r r<rAs r1_eval_rewrite_as_tanhzexp._eval_rewrite_as_tanh2s(>DQK!d3q5k/22r2c ddlm}m}|jrq|jt t z}|rQ|jrD|t |z|t |z}}t||st||s |t |zzSyyyyy)Nr)rr) rrrrrrrrrt)r0r r<rrrcosinesines r1_eval_rewrite_as_sqrtzexp._eval_rewrite_as_sqrt6swE ::CIIbdOE"2e8}c"U(m!&#.z47M!AdF?*8N.)u r2c |jru|jDcgc].}t|tst |jdk(s-|0}}|r/t |djd|j |dSyycc}wNrDr)rrBrtr5lenrr)r0r r<rlogss r1_eval_rewrite_as_Powzexp._eval_rewrite_as_Pow?sq ::"xxS!:a+=#aff+QRBRASDS47<<?ICIId1g,>?? SsBB BryTrr)rzr{r|rrr classmethodrrr staticmethodrrrrrrrrrr%r:r=r?rCrGrLrr2r1r-r-s45!(ggR   !  !*@ 3 A  +Z>*003+@r2r-) metaclassc|jtd\}}|dk(r|jr||fS|jt}|r|jr|jr||fSy)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 r1match_real_imagrZFs^ 4 0FB Qw2::Bx  1 B bjjRZZBxr2ceZdZUdZeeed<ejejfZ ddZ ddZ e ddZdZeedZdd Zd Zdd Zd Zd ZdZdZdZdZdZdZddZddZy)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 rBcH|dk(rd|jdz St||)z? Returns the first derivative of the function. rDr)rBr r6s r1rz log.fdiffs* q=TYYq\> !$T84 4r2ctS)zC Returns `e^x`, the inverse function of `\log(x)`. )r-r6s r1r8z log.inverser9r2Ncddlm}ddlm}t |}|{t |}|dk(r%|dk(rt j St jS t||}|r |t|||zz t|z zSt|t|z S|jr|jrt jS|t jurt jS|t j urt j S|t j"urt j S|t j urt j S|j$r"|j&dk(r||j( S|j*r>|j,t jur"|j.j0r |j.St3|t.r"|j.j0r |j.St3|t.rv|j.j4r`t7|j.\}}|r)|j8r|dt:zz}|t:kDr |dt:zz}|t=|t>zdzSt3|t@rtC|j.St3||r|jDjFr/|t|jDt|jHS|jDjr*|t j"t|jHSt j St3||r|jJ|S|j4rg|jLrt:t>z|| zS|t jurt jS|t jurt jS|jrt jS|jNs|jPt>}||t j urt j S|t j"urt j S|j$r\|jRr't:t>zt jTz||zSt: t>zt jTz|| zS|j4r|jVr|jXt>d\}} |jLr |d z}| d z} t=| d} | jYt>d \}}|jQt>}|jZr{|rw|jZri|jZr[|jro|jFr*t:t>zt jTz|||zzS|jLr,t: t>zt jTz||| zzSydd l.m/} ||z ja} | ja} tc} | | vrQ| |te| z}|jFr||t>| | zzS||t>| | t:z zzS| | vrR| |te| z}|jFr||t>| |  zzS||t>t:| | z zzSyyyyyyy#t$rYnwxYw|t jur||||z S||S) NrrrrDrFrrSrT)ratsimp)3rrrrrrrr~r&r5 ValueErrorrrrYr=rrr_rrrrrr-rrtrrZrrr rrr!rrrrr;rqris_nonnegativerrrUrVsympy.simplifyr`r9_log_atan_tabler$)rr rrrrrXrYrarg_r`rt1 atan_tablemoduluss r1rzlog.evals..cl  4=Dqy!855L,,, !s+s3q=1CI===s8CI-- =={{(((vv  "zz!***zz!uu SUUaZCEE {" ::#((aff,1I1I77N c3 CGG$<$<77N S !cgg&7&7$SWW-FBb&&ad 7!B$JBJrAvE::: Y 'cgg& & [ )ww"""3sww<SWW>>"1#5#5s377|DDuu W %!3>>#& & ==AvSD )))))(((uu ;;$$ $zz&C&&q)E AJJ&::%a000::%&&++!AvU;; "sQw/#uf+== ==S--,#,,Qu=KE4    d/D((4(8FB""1%B}} rzz::~~!AvURZ@@ "sQw/#erck2BBB(7B(A"B!0!2JJ")%#d)*;"<>>#&w>#&w[ [[c&|tjfS)zE Returns this function in the form (base, exponent). )rr=r/s r1rEzlog.as_base_expsQUU{r2cddlm}|dkrtjSt |}|dk(r|S|r|d}||| |z|z|dzz ddSdd|dzzz ||dzzz|dzz S) zV Returns the next term in the Taylor series expansion of `\log(1+x)`. rr rrDTr-rr)rr rrr)rrmrr rs r1rzlog.taylor_terms 3 q566M AJ 6H r"A}ax!|q1u5D%PPAq1uI QU+QU33r2c 0ddlm}m}|jdd}|jdd}t |j dk(r%t |j|j ||S|j d}|jrpt|}d} d} |dur|\}} |j|} |r=t|}||jvr td |jD} | | | zS|jr+t|j t|j"z S|j$r g} g} |j D]} |s| j&s | j(rd|j| }t+|tr1| j-|j| j.d i|o| j-|| j0rC|j| }| j-|| j-t2j4| j-| t7| tt9| zS|j:st+|t<r|st|j<j>rH|j@j&sH|j<dzj&r|j<dz jBs|j@j(r|j@}|j<}|j|}t+|trtE||j.d i|zStE||zSt+||r>|s|jFj&r&|t|jFg|jHS|j|S) Nr)rjriforceFfactorr)rrlrDc3>K|]\}}|t|zywr,r4)rlvalrs r1rnz'log._eval_expand_log..9s DQ3s8 Dsr)%sympy.concreterjrigetrJrBr r> is_Integerr'r(keyssumitemsrr5rrrrrrtr_eval_expand_logr;rrrrrr-rris_nonpositiver!rurv)r0rrwrjrirlrmr rlogargrrWnonposrmrrfrgs r1rvzlog._eval_expand_log&s/ '5)8U+  Na idii3$eL Liil >>c"AFE~ U3cNaffh& D!'') 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"1? 7*:s$j*r2r5ceZdZdZeej dd ejfZe d dZ ddZ dZ dZ d Zdd Zdfd Zd ZxZS)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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