\c@ spddlmZmZddlmZddlmZddlmZddl m Z ddl m Z m Z mZddlmZddlmZdd lmZdd lmZdd lmZdd lmZmZdd lmZddlmZm Z de fdYZ!de!fdYZ"de!fdYZ#de fdYZ$de fdYZ%dS(i(tprint_functiontdivision(tsympify(tAdd(tcacheit(trange(tFunctiontArgumentIndexErrort _coeff_isneg(t fuzzy_not(tMul(tInteger(tPow(tS(tWildtDummy(t factorial(t multiplicityt perfect_powertExpBasecB sqeZeZddZdZedZdZdZ dZ dZ dZ d Z d ZRS( icC stS(s= Returns the inverse function of ``exp(x)``. (tlog(tselftargindex((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pytinverse!scC s]|j}|j}| r3| j r3t|}n|rPtj|j| fS|tjfS(s7 Returns this with a positive exponent as a 2-tuple (a fraction). Examples ======== >>> from sympy.functions import exp >>> from sympy.abc import x >>> exp(-x).as_numer_denom() (1, exp(x)) >>> exp(x).as_numer_denom() (exp(x), 1) (texpt is_negativeRR tOnetfunc(RRtneg_exp((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pytas_numer_denom's  cC s |jdS(s7 Returns the exponent of the function. i(targs(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR?scC s|jdt|jfS(s7 Returns the 2-tuple (base, exponent). i(RR R(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt as_base_expFscC s|j|jdjS(Ni(RRt conjugate(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_conjugateLscC sD|jd}|jr3|jr#tS|jr3tSn|jr@tSdS(Ni(Rt is_infiniteRtTruet is_positivetFalset is_finite(Rtarg((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_is_finiteOs     cC sj|j|j}|j|jkr_|jtjkr:tS|jjrft|jjrft Sn|jSdS(N( RRRR tZeroR#t is_rationalR tis_zeroR%(Rts((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_is_rationalYscC s|jdtjkS(Ni(RR tNegativeInfinity(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt _eval_is_zerocscC s1|j\}}tjt||dt|S(s;exp(arg)**e -> exp(arg*e) if assumptions allow it. tevaluate(RR t _eval_powerR%(Rtothertbte((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR1fscK s]|jd}|jrP|jrPd}x$|jD]}||j|9}q/W|S|j|S(Nii(Rtis_Addtis_commutativeR(RthintsR'texprtx((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_expand_power_expls (t__name__t __module__R#t unbranchedRRtpropertyRRR!R(R-R/R1R:(((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs      t exp_polarcB sGeZdZeZeZdZdZdZ dZ dZ RS(so Represent a 'polar number' (see g-function Sphinx documentation). ``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. >>> 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.simplify.powsimp sympy.functions.elementary.complexes.polar_lift sympy.functions.elementary.complexes.periodic_argument sympy.functions.elementary.complexes.principal_branch cC s'ddlm}t||jdS(Ni(trei(t$sympy.functions.elementary.complexesR@RR(RR@((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt _eval_AbsscC sddlm}m}m}||jd}y|| kpH||k}Wntk ret}nX|rp|St|jdj|}|dkr||dkr||S|S(s. Careful! any evalf of polar numbers is flaky i(timtpiR@i( tsympyRCRDR@Rt TypeErrorR#Rt _eval_evalf(RtprecRCRDR@titbadtres((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRGs   cC s|j|jd|S(Ni(RR(RR2((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR1scC s|jdjrtSdS(Ni(Rtis_realR#(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt _eval_is_realscC s0|jddkr#|tdfStj|S(Nii(RR RR(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs( R;R<t__doc__R#tis_polarR%t is_comparableRBRGR1RMR(((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR?vs    RcB seZdZddZdZedZedZe e dZ e dZ dZd Zd Zd Zd Zd ZdZdZdZdZdZdZRS(sT The exponential function, :math:`e^x`. See Also ======== log icC s#|dkr|St||dS(s@ Returns the first derivative of this function. iN(R(RR((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pytfdiffs cC sddlm}m}|jd}|jrtjtj}||| gkrVtjS|j tj tj}|r||j d|r||j |rtj S||j|rtjS||j |tjrtj S||j|tjrtjSqqndS(Ni(tasktQii(tsympy.assumptionsRRRSRtis_MulR t ImaginaryUnittInfinitytNaNtas_coefficienttPitintegertevenRtoddt NegativeOnetHalf(Rt assumptionsRRRSR'tIootcoeff((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt _eval_refines"  cC sddlm}m}ddlm}ddlm}ddlm}ddl m }|j r|t j krut j S|t jkrt jS|t jkrt jS|t jkrt jS|t jkrt jSn|t jkrt j St|tr|jdSt||r.|t|jt|jSt||rJ|j|S|jr|jse|jr%|jt jt j }|r%||j!d|r"||j"|rt jS||j#|rt j$S||j"|t j%rt j S||j#|t j%rt j Sq"q%n|j&\}} |t jt jgkrSdS|gd} } xrt(j)| D]a} || } t| tr| dkr| jd} qdSqs| j*r| j+| qsdSqsW| r| t(| SdS|j,rg}g}xk|jD]`}|t jkr7|j+|qn||}t||rb|j+|q|j+|qW|rt(||t-|d t.Snt||r|jSdS( Ni(RRRS(t AccumBounds(tSetExpr(t MatrixBase(t logcombineiiR0(/RTRRRStsympy.calculusRdtsympy.sets.setexprRetsympy.matrices.matricesRfRERgt is_NumberR RXR)RtExp1RWR.tComplexInfinityt isinstanceRRRtmintmaxt _eval_funcRUt is_numbert is_SymbolRbRZRVR[R\R]R^R_t as_coeff_MultNoneR t make_argsRPtappendR5RR%(tclsR'RRRSRdReRfRgRbttermstcoeffstlog_termttermtterm_touttaddtatnewa((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pytevals           #cC stjS(s? Returns the base of the exponential function. (R Rl(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pytbase6scG so|dkrtjS|dkr&tjSt|}|r]|d}|dk r]|||Sn||t|S(sJ Calculates the next term in the Taylor series expansion. iiN(R R)RRRuR(tnR9tprevious_termstp((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt taylor_term=s     cK sddl}|jdj\}}|rR|j||}|j||}n|j||j|}}t||t||fS(sq 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 iNi(RERt as_real_imagtexpandtcostsinR(RtdeepR7RER@RCRR((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRNs cC s|jr(t|jt|j}n!|tjkrI|jrIt}nt|tsg|tjkrd}tj |||||S|tkr|j r||jj ||St j |||S(NcS s2|jst|tr.tdt|jS|S(NR0(tis_PowRnRR R%R(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pytvs( RRRRR Rlt is_FunctionRnR t _eval_subst_subsR(Rtoldtnewtf((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRos   cC sU|jdjrtS|jdjrQtd tj|jdtj}|jSdS(Nii(RRLR#t is_imaginaryR RVRZtis_even(Rtarg2((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRM~s &cC sn|j|j}|j|jkrct|jjrj|jjrFtS|jtjj r`tSqjn|jSdS(N( RRR RR+t is_algebraicR%R RZR*(RR,((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_is_algebraics  cC s[|jdjr$|jdtjk S|jdjrWtj |jdtj}|jSdS(Ni(RRLR R.RRVRZR(RR((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_is_positives cC s.ddlm}m}m}m}|jd}|j|d|d|} | jr[d| S|| j|d} | | |gkr|St d} t | j | |} | j } | j} t | | j | | | }||| jj | | | |7}|j}||dtd d S( Ni(tlimittootOrdertpowsimpiRtlogxittRtcombineR(RERRRRRt _eval_nseriestis_OrdertremoveORRt_taylortgetOtsubsR8RR#(RR9RRRRRRR't arg_seriestarg0Rt exp_seriestotr((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs "      & cC sddlm}g}d}xRt|D]D}|j||jd|}|j|d|}|j|q)Wt|||||S(Ni(RiR( RERRuRRRtnseriesRwR(RR9RRtltgRI((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRscC sddlm}|jd}|jrUtg|jD]}t|j|^q3S|jdj|}|d|j|rtj St|S(Ni(Rii( RERRR5R Rtas_leading_termtcontainsR R(RR9RR'R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_as_leading_terms  /cK sDddlm}tj}|||tjd||||S(Ni(Ri(RERR RVRZ(RR'tkwargsRtI((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_rewrite_as_sins cK sDddlm}tj}|||||||tjdS(Ni(Ri(RERR RVRZ(RR'RRR((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_rewrite_as_coss cK s4ddlm}d||dd||dS(Ni(ttanhii(RER(RR'RR((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_rewrite_as_tanhscK sddlm}m}|jr|jtjtj}|r|jr|tj||tj|}}t || rt || r|tj|SqndS(Ni(RR( t(sympy.functions.elementary.trigonometricRRRURbR RZRVRrRn(RR'RRRRbtcosinetsine((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_rewrite_as_sqrts ' cK s{|jrwg|jD]0}t|trt|jdkr|^q}|rwt|djd|j|dSndS(Nii(RURRnRtlenR Rb(RR'RRtlogs((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_rewrite_as_Pows @(R;R<RNRQRct classmethodRR>Rt staticmethodRRR#RRRMRRRRRRRRRR(((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs( T !        RcB seZdZddZddZeddZdZe e dZ e dZ dZe d Zd Zd Zd Zd ZdZdZdZdZdZRS(s The natural logarithm function `\ln(x)` or `\log(x)`. 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)``. See Also ======== exp icC s.|dkrd|jdSt||dS(s? Returns the first derivative of the function. iiN(RR(RR((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRQs cC stS(sC Returns `e^x`, the inverse function of `\log(x)`. (R(RR((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRsc C sddlm}ddlm}ddlm}t|}|dk r<t|}|dkr}|dkrstj Stj Sny{t ||}|r||}|j r|t ||t |S|t ||t |Snt |t |SWntk r nX|tjk r/||||S||Sn|jr|tjkr[tj S|tjkrqtjS|tjkrtjS|tjkrtjS|tj krtj S|jr|jdkr||j Sn|tj krtj St|tr|jdjr|jdSt|tr8||jSt||ry|jjrr|t |jt |jSdSnt||r|j |S|j!r|j"rtj#tj$|| S|tj krtj S|tjkrtjSn|j%s|j&tj$}|dk r|tjkr,tjS|tjkrBtjS|jr|j'rstj#tj$tj(||Stj# tj$tj(|| SqqndS(Ni(t unpolarify(Rd(Reii()RERRhRdRiReRRuR RXRmRt is_IntegerRt ValueErrorRlRkR)RRWR.t is_RationalRtqRnRRRLR?RoR$RpRqRrRRZRVR5RYtis_nonnegativeR_( RxR'RRRdReRtdenRb((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs                      cC s |tjfS(sE Returns this function in the form (base, exponent). (R R(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRNscG sddlm}|dkr#tjSt|}|dkr?|S|r|d}|d k r|| |||ddtddSndd|d||d|dS( sV Returns the next term in the Taylor series expansion of `\log(1+x)`. i(RiiRRRiN(RERR R)RRuR#(RR9RRR((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRTs     *cK sddlm}m}ddlm}m}|jdt}t|j dkru||j |j d|d|S|j d}|j rt t |} | tk r| d|j | dSn|jrt|jt|jS|jrg} g} x|j D]} |s%| js%| jru|j | } t| tre| j|j | j|q| j| q| jr|j | } | j| | jtjq| j| qWt| tt| S|jst|tr|s;|jjr/|j js;|jdjr/|jdj!s;|j jr|j }|j}|j |} t| tr||| j|S||| Sqn7t||r|j"jr|t|j"|j#Sn|j |S( Ni(Rt expand_log(tSumtProducttforceiRii($RERRtsympy.concreteRRtgetR%RRRRRtintRRRRRUR$RORnRwt_eval_expand_logRR R^RR RRRLRtis_nonpositivetfunctiontlimits(RRR7RRRRRR'RR8tnonposR9RR3R4((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRfsP"     "  .   c C sddlm}m}m}t|jdkr_||j|jd|d|d|d|S|j||jdd|d|d|d|}|r||}n||d t}t||gd |S( Ni(RtsimplifytinversecombineitratiotmeasuretrationalRiRtkey( tsympy.simplify.simplifyRRRRRRR#Ro( RRRRRRRRR8((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_simplifys! "cK sddlm}m}|r]||jdj||}||jdj||}n&||jd}||jd}|jdtrt|d>> 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)) i(tAbsR'iRtcomplexN(RERR'RRRR%R(RRR7RR'tabs((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs" cC sx|j|j}|j|jkrm|jddjr<tS|jdjrtt|jddjrttSn|jSdS(Nii(RRR+R#R*R R%(RR,((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR-s*cC s{|j|j}|j|jkrp|jddjr<tSt|jddjrw|jdjrmtSqwn|jSdS(Nii(RRR+R#R RR%(RR,((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs cC s|jdjS(Ni(RR$(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRMscC s!|jd}|jrtS|jS(Ni(RR+R%R&(RR'((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR(s  cC s|jddjS(Nii(RR$(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRscC s|jddjS(Nii(RR+(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR/scC s|jddjS(Nii(RR(R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyt_eval_is_nonnegativescC sddlm}m}|s+t|}n|jd|krB|S|jd}tdtd}}|j|||} | dk r| || |}}|dkr|j| r|j| rt|||} | Sn|jdj |d|d|} x9| j rD|d7}|jdj |d|d|} q W| j |\} } || | || d} d}g}xUt |d D]C}tj || |}|j |d|d|}|j|qWt| | |t||| ||S( Ni(tcancelRitkRRRii(RERRRRRtmatchRuthasRRtleadtermRRRwR(RR9RRRRR'RRRR,RR3RRRI((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs4  ,"  &cC sJ|jdj|}|tjkr=|jddj|S|j|S(Nii(RRR RR(RR9R'((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR sN(R;R<RNRQRRRuRRRRRR#RRRR-RRMR(RR/RRR(((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs(  T  / !      tLambertWcB s>eZdZeddZddZdZdZRS(s The Lambert W function `W(z)` is defined as the inverse function of `w \exp(w)` [1]_. 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 ========== .. [1] https://en.wikipedia.org/wiki/Lambert_W_function cC sa|tjkr||S|dkr1tj}n|tjkr|tjkrVtjS|tjkrltjS|dtjkrtjS|td dkrtd S|tjkrtjSnt|j r|tjkrtj Sn|tjkr]|tj dkrtj tj dS|dtjkr9tjS|dt dkr]td SndS(Niii(R R)RuRlRR^RRWR R+R.RZRVRR (RxR9R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyR1s2      icC s|jd}t|jdkrM|dkrt||dt|Sn;|jd}|dkrt|||dt||St||dS(s? Return the first derivative of this function. iiN(RRRR(RRR9R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRQOs    "cC s|jd}t|jdkr.tj}n |jd}|jrw|dtjjr\tS|dtjjrt Sn|djr|j r|dtjjrtS|js|dtjj rt Sn2t |jrt |djr|j rt SndS(Nii(RRR R)R+RlR$R#RR%RRR RL(RR9R((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRM_s"     " cC s\|j|j}|j|jkrQt|jdjrX|jdjrXtSn|jSdS(Ni(RRR R+RR%(RR,((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRss &N( R;R<RNRRuRRQRMR(((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyRs   N(&t __future__RRt sympy.coreRtsympy.core.addRtsympy.core.cacheRtsympy.core.compatibilityRtsympy.core.functionRRRtsympy.core.logicR tsympy.core.mulR tsympy.core.numbersR tsympy.core.powerR tsympy.core.singletonR tsympy.core.symbolRRt(sympy.functions.combinatorial.factorialsRt sympy.ntheoryRRRR?RRR(((sElib/python2.7/site-packages/sympy/functions/elementary/exponential.pyts(YE#4