~9\c@ sdZddlmZmZddlmZddlmZmZm Z ddl m Z ddl m Z ddlmZddlmZmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZmZm Z m!Z!ddl"m#Z#ddl$m%Z%m&Z&m'Z'ddl(m)Z)m*Z*m+Z+ddl,m-Z-m.Z.ddl/m0Z0ddl1m2Z2ddl3m4Z4ddl5m6Z6ddl7m8Z8de9dZ:dZ;ddZ<dZ=dZ>dZ?dZ@d ZAd!ZBd"ZCd#ZDd$ZEd%ZFd&ZGd'ZHd(ZId)ZJd*ZKd+ZLd,ZMdd-ZNd.ZOd/d0ePdePe9d1ZQd2e8fd3YZReSd/d0ePdePe9d4ZTd5S(6sFormal Power Seriesi(tprint_functiontdivision(t defaultdict(tootzootnan(tAdd(titerable(tExpr(t DerivativetFunction(tMul(tRational(tEq(tInterval(tS(tWildtDummytsymbolstSymbol(tsympify(tbinomialt factorialtrf(tfloortfractceiling(tMintMax(t Piecewise(tLimit(tOrder(tsequence(t SeriesBaseicC sddlm}m}ddlm}|}g} xt|dD]t} | ra|j|}n|j|rtj tj } } |||d|} | j |r| j } nx)t j | D]}|j\}}|j |s| |7} qt|tr3|j|}|d}||d}n|j\}}|j|\}}|sp| |7} qn|dj|}|| }|||}d||t||d|jt|||}| |7} qW| tj krdS| j |s1| j ts1| j ts1| j tr5dSxUt| D]G}| ||d} || |} | | j| j|d7} qBW| j||| | | fS| j|qCWdS(s?Rational algorithm for computing formula of coefficients of Formal Power Series of a function. Applicable when f(x) or some derivative of f(x) is a rational function in x. :func:`rational_algorithm` uses :func:`apart` function for partial fraction decomposition. :func:`apart` by default uses 'undetermined coefficients method'. By setting ``full=True``, 'Bronstein's algorithm' can be used instead. Looks for derivative of a function up to 4'th order (by default). This can be overridden using order option. Returns ======= formula : Expr ind : Expr Independent terms. order : int Examples ======== >>> from sympy import log, atan, I >>> from sympy.series.formal import rational_algorithm as ra >>> from sympy.abc import x, k >>> ra(1 / (1 - x), x, k) (1, 0, 0) >>> ra(log(1 + x), x, k) (-(-1)**(-k)/k, 0, 1) >>> ra(atan(x), x, k, full=True) ((-I*(-I)**(-k)/2 + I*I**(-k)/2)/k, 0, 1) Notes ===== By setting ``full=True``, range of admissible functions to be solved using ``rational_algorithm`` can be increased. This option should be used carefully as it can significantly slow down the computation as ``doit`` is performed on the :class:`RootSum` object returned by the ``apart`` function. Use ``full=False`` whenever possible. See Also ======== sympy.polys.partfrac.apart References ========== .. [1] Formal Power Series - Dominik Gruntz, Wolfram Koepf .. [2] Power Series in Computer Algebra - Wolfram Koepf i(tRootSumtapart(t integrateitfulliN( t sympy.polysR"R#tsympy.integralsR$trangetdifftis_rational_functionRtZerothastdoitRt make_argstas_numer_denomt isinstanceR tas_independentt as_base_expt as_coeff_addtcoeffRtrewriteRtNoneRRRtpoptlimittsubstappend(tftxtktorderR%R"R#R$R)tdstiR4tsepttermstttnumtdentindtjtatxtermtxctak((s2lib/python2.7/site-packages/sympy/series/formal.pytrational_algorithmsV;    )-$c C s|s gS|dd!}x|dD]}|j|d}xmt|D]R\}}|j|d}||j}|j|rH||c|7>> from sympy import sin, cos >>> from sympy.series.formal import rational_independent >>> from sympy.abc import x >>> rational_independent([cos(x), sin(x)], x) [cos(x), sin(x)] >>> rational_independent([x**2, sin(x), x*sin(x), x**3], x) [x**3 + x**2, x*sin(x) + sin(x)] ii(R1t enumeratetcancelR*R:( RBR<RFRCtnR@ttermtdtq((s2lib/python2.7/site-packages/sympy/series/formal.pytrational_independents c # s]ddlm}td|fd}||\}}t}xtd|dD]} || \}}|j}|j} t| } |st| | krdt t d|| | D} | rt }|j | }n|j d}|jjtdd}|jt| fVqdqdWdS( s^Generates simple DE. DE is of the form .. math:: f^k(x) + \sum\limits_{j=0}^{k-1} A_j f^j(x) = 0 where :math:`A_j` should be rational function in x. Generates DE's upto order 4 (default). DE's can also have free parameters. By increasing order, higher order DE's can be found. Yields a tuple of (DE, order). i(tlinsolvesa:%dc sj|tgtd|D] }|j|^q"}j|tgtd|D]&}|j|^qt}||fS(Ni(R)RR((R=R@teqtDE(RHR;tgR<(s2lib/python2.7/site-packages/sympy/series/formal.pyt_makeDEsLXics s"|]}|D] }|Vq qdS(N((t.0tsR@((s2lib/python2.7/site-packages/sympy/series/formal.pys siN(tsympy.solvers.solvesetRTRtFalseR(texpandtas_ordered_termsRStlentdicttziptTrueR9R/tfactort as_coeff_mulR tcollect( R;R<RWR>RTRXRURVtfoundR=RBRFtsol((RHR;RWR<s2lib/python2.7/site-packages/sympy/series/formal.pytsimpleDEs$  ,c C stj}|jtj}d}xtj|D]u}|j|\}}t |t rj|j } nd} |dks| |kr| }n||||| 7}q4W|r|j |||}n|S(szConverts a DE with constant coefficients (explike) into a RE. Performs the substitution: .. math:: f^j(x) \to r(k + j) Normalises the terms so that lowest order of a term is always r(k). Examples ======== >>> from sympy import Function, Derivative >>> from sympy.series.formal import exp_re >>> from sympy.abc import x, k >>> f, r = Function('f'), Function('r') >>> exp_re(-f(x) + Derivative(f(x)), r, k) -r(k) + r(k + 1) >>> exp_re(Derivative(f(x), x) + Derivative(f(x), (x, 2)), r, k) r(k) + r(k + 1) See Also ======== sympy.series.formal.hyper_re iN( RR+tatomsR R7R6RR.R1R0R tderivative_countR9( RVtrR=tRERWtminiRCR4RQRG((s2lib/python2.7/site-packages/sympy/series/formal.pytexp_res   cC sJtj}|jtj}|jtj}d}xtj|j D]}|j |\}} |j |\} } | j |d} t | t r| j} nd} || t|d| | ||| | 7}|dks| | |krO| | }qOqOW|j|||}td}|j|||S(s}Converts a DE into a RE. Performs the substitution: .. math:: x^l f^j(x) \to (k + 1 - l)_j . a_{k + j - l} Normalises the terms so that lowest order of a term is always r(k). Examples ======== >>> from sympy import Function, Derivative >>> from sympy.series.formal import hyper_re >>> from sympy.abc import x, k >>> f, r = Function('f'), Function('r') >>> hyper_re(-f(x) + Derivative(f(x)), r, k) (k + 1)*r(k + 1) - r(k) >>> hyper_re(-x*f(x) + Derivative(f(x), (x, 2)), r, k) (k + 2)*(k + 3)*r(k + 3) - r(k) See Also ======== sympy.series.formal.exp_re iitmN(RR+RiR R7RR6RR.R]R1tas_coeff_exponentR0R RjRR9RRe(RVRkR=RlRWR<RmRCR4RQtctvtlRGRo((s2lib/python2.7/site-packages/sympy/series/formal.pythyper_res"  1 cC sK||| 9}|j|||}|j|||}||||fS(N(R9(R;R<tPtQR=Rotshift((s2lib/python2.7/site-packages/sympy/series/formal.pyt_transformation_aHscC s\|j|||}|j|||}|j|||}||9}||||fS(N(R9(R;R<RuRvR=Rotscale((s2lib/python2.7/site-packages/sympy/series/formal.pyt_transformation_cOs  cC s_|j|}|j||d||d}|j||d|d}||||fS(Ni(R)R9(R;R<RuRvR=Ro((s2lib/python2.7/site-packages/sympy/series/formal.pyt_transformation_eWs"cC s'g|D]\}}|||f^qS(N((RgRwtrestcond((s2lib/python2.7/site-packages/sympy/series/formal.pyt _apply_shift^scC s'g|D]\}}|||f^qS(N((RgRyR|R}((s2lib/python2.7/site-packages/sympy/series/formal.pyt _apply_scalebscC sFg|D];\}}||d|jdj||df^qS(Ni(t as_coeff_AddR4(RgR<R=R|R}((s2lib/python2.7/site-packages/sympy/series/formal.pyt_apply_integratefscC sddlm}g}xt|d||dD]{} | dktkrPq2n|j|| j|dt| } | tjkrq2n||| } | } |j || } |j || }| j |d|j |d}|j |d|j |d}| | ||9} x;|| |j D]$\} }| t | ||9} q4Wx;|||j D]$\} }| t | ||} qrW|j | | fq2W|S(sComputes the formula for f.i(trootsii(R&RR(RbR)R8RRR+R9tleadtermtitemsRR:(R;R<RuRvR=Rotk_maxRRgR@RktktermR|tpRRtc1tc2tmul((s2lib/python2.7/site-packages/sympy/series/formal.pyt_compute_formulaks*"(##""c C sddlm}m}ddlm}||||||} } t| } | j| |g| jD]%\} } | jrn| j d^qn}t |||||||\}}}}|||} | rt | j }n t j}||}t|||||||\}}}}||j|d}t|t rc|dkrcdS|||} | rt| j }n t j}t jt }}xvt||dD]`}|j||j|dt|} | jtkr|}t|||||||\}}}}t||||||\}}}}t||||||\}}}t|||}t||}|||}|||j|d7}|d7}|||fS| r|| |||7}t|||}||kr|}qqqW|j ||d|}t!|||||||}t||}t"||}|||fS(sRecursive wrapper to rsolve_hypergeometric. Returns a Tuple of (formula, series independent terms, maximum power of x in independent terms) if successful otherwise ``None``. See :func:`rsolve_hypergeometric` for details. i(tlcmR(R$iiN(#R&RRR'R$R`tupdateRt is_rationalR/RzRtkeysRR+RxR8R0RR6RRR(R)Rt is_finiteR\R{t_rsolve_hypergeometricRR~R R9RR(R;R<RuRvR=RoRRR$tprootstqrootst all_rootsRkRCRytk_minRwRsRRFtmpR@told_fRgtpow_x((s2lib/python2.7/site-packages/sympy/series/formal.pyRsZ   "*  * (*'$   cC st||||||}|dkr+dS|\}}} td} x|D]\} } | j\} }|j|}| jtkr| |t| 9} t| } n| j ||| |} t ||| |} | | c| 7>> from sympy import exp, ln, S >>> from sympy.series.formal import rsolve_hypergeometric as rh >>> from sympy.abc import x, k >>> rh(exp(x), x, -S.One, (k + 1), k, 1) (Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1) >>> rh(ln(1 + x), x, k**2, k*(k + 1), k, 1) (Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1), Eq(Mod(k, 1), 0)), (0, True)), x, 2) References ========== .. [1] Formal Power Series - Dominik Gruntz, Wolfram Koepf .. [2] Power Series in Computer Algebra - Wolfram Koepf cS stjS(N(RR+(((s2lib/python2.7/site-packages/sympy/series/formal.pyttiiiN(RR6RRR4t is_integerR\RRR9R RR:RR+RbRRRtsumR (R;R<RuRvR=Rotresulttsol_listRFRtsol_dictR|R}RGtmkRqRgRZ((s2lib/python2.7/site-packages/sympy/series/formal.pytrsolve_hypergeometrics8-      * c C stj|}t|dkrt|jt}t|j|\}}|djd|djd} | dkr||}}t | } nt |||||| SdS(s;See docstring of :func:`rsolve_hypergeometric` for details.iiiN( RR.R_tlistRiR tmapR4targstabsR( R;R<RlRWR=RBtgsRuRvRo((s2lib/python2.7/site-packages/sympy/series/formal.pyt_solve_hyper_RE$s   c C s ddlm}x9tj|D](}|j|\}}|jr dSq Wt|||} i} xctttj| D]F} | r|j |}n|j |d| ||j || osii(R[RTR4R R(R]ReRR.R:R,RR`RaRR9RcRd(RVRWR<R>tsymsRTRUt highest_coeffR@R4RCttempteRg((s2lib/python2.7/site-packages/sympy/series/formal.pyt_transform_explike_DE[s,!! .!c C sNddlm}t|||}g}x=td|D],}|j|||} |j| q8Wtt|d||t|D} | rJt d} |j | }|j j dj ||| }|j|dd}xPt|D]?}|j|||r|r|j |||}PqqWn|S(s@Converts DE with free parameters into RE of hypergeometric type.i(RTics s"|]}|D] }|Vq qdS(N((RYRZR@((s2lib/python2.7/site-packages/sympy/series/formal.pys sRoi(R[RTRtR(R4R:R`RaRRR9RcR/ReRd( RVRWR=R>RRTRlRUR@R4RgRo((s2lib/python2.7/site-packages/sympy/series/formal.pyt_transform_DE_REws". ) c C sd}|jj||h}|r?t|||||}nt|||}|jj|hst|||||}n|r|S|rt|||||}n|jj|hst|||||}n|r|SdS(sSolves the DE. Tries to solve DE by either converting into a RE containing two terms or converting into a DE having constant coefficients. Returns ======= formula : Expr ind : Expr Independent terms. order : int Examples ======== >>> from sympy import Derivative as D, Function >>> from sympy import exp, ln >>> from sympy.series.formal import solve_de >>> from sympy.abc import x, k >>> f = Function('f') >>> solve_de(exp(x), x, D(f(x), x) - f(x), 1, f, k) (Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1) >>> solve_de(ln(1 + x), x, (x + 1)*D(f(x), x, 2) + D(f(x)), 2, f, k) (Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1), Eq(Mod(k, 1), 0)), (0, True)), x, 2) N(R6Rt differenceRRtRRR( R;R<RVR>RWR=RgRRl((s2lib/python2.7/site-packages/sympy/series/formal.pytsolve_desc C std}g}d}x|t||||D]e\}}|dk rdt||||||}n|rn|S|jj|hs.|j|q.q.Wx0|D](}t|||||}|r|SqWdS(sHypergeometric algorithm for computing Formal Power Series. Steps: * Generates DE * Convert the DE into RE * Solves the RE Examples ======== >>> from sympy import exp, ln >>> from sympy.series.formal import hyper_algorithm >>> from sympy.abc import x, k >>> hyper_algorithm(exp(x), x, k) (Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1) >>> hyper_algorithm(ln(1 + x), x, k) (Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1), Eq(Mod(k, 1), 0)), (0, True)), x, 2) See Also ======== sympy.series.formal.simpleDE sympy.series.formal.solve_de RWN(R R6RhRRRR:R( R;R<R=R>RWtdesRgRVR@((s2lib/python2.7/site-packages/sympy/series/formal.pythyper_algorithms "  c C s|tjtj gkr|tjkr1tjntj }|j|d|}t||d|||||} | dkrdS| d| dj|d|| dj|d|fS|s|tj kr|tj kr| |} | } |} n||} |} | } |j|| }t||dtj||||} | dkrYdS| d| dj|| | | dj|| | fS|j|rdSt|tr t } t tj dt f} tj d}}x$tj |D]}t||dtj||||}|r| s;t} |d}n|dj| jkrq| }| j|dj}}n!|d}|dj| j}}tgt|d||!|||!D]}|d|d^q}| |d7} ||d|7}q||7}qW| r| ||fSdSd} td}|rSt|||||} n| dkr}|r}t||||} n| dkrdSt | d|| dt f} t |||dt f}| d}| ||fS(sPRecursive wrapper to compute fps. See :func:`compute_fps` for details. iiiR=N(RtInfinitytOneR9t _compute_fpsR6t is_polynomialR0RR\R R+RR.RbtstartRaRRLR(R;R<tx0tdirthyperR>trationalR%RRtreptrep2trep2bRKRFtxkRCR|tseqRZtztsaveR=((s2lib/python2.7/site-packages/sympy/series/formal.pyRsr"! 9   $ $  F     iic C st|}t|}|j|s+dSt|}|dkrOtj}nM|dkrhtj }n4|tjtj gkrtdn t|}t||||||||S(sComputes the formula for Formal Power Series of a function. Tries to compute the formula by applying the following techniques (in order): * rational_algorithm * Hypergeomitric algorithm Parameters ========== x : Symbol x0 : number, optional Point to perform series expansion about. Default is 0. dir : {1, -1, '+', '-'}, optional If dir is 1 or '+' the series is calculated from the right and for -1 or '-' the series is calculated from the left. For smooth functions this flag will not alter the results. Default is 1. hyper : {True, False}, optional Set hyper to False to skip the hypergeometric algorithm. By default it is set to False. order : int, optional Order of the derivative of ``f``, Default is 4. rational : {True, False}, optional Set rational to False to skip rational algorithm. By default it is set to True. full : {True, False}, optional Set full to True to increase the range of rational algorithm. See :func:`rational_algorithm` for details. By default it is set to False. Returns ======= ak : sequence Sequence of coefficients. xk : sequence Sequence of powers of x. ind : Expr Independent terms. mul : Pow Common terms. See Also ======== sympy.series.formal.rational_algorithm sympy.series.formal.hyper_algorithm t+t-sDir must be '+' or '-'N(RR,R6RRt ValueErrorR(R;R<RRRR>RR%((s2lib/python2.7/site-packages/sympy/series/formal.pyt compute_fpsCs3        tFormalPowerSeriescB s[eZdZdZedZedZedZedZedZ edZ edZ ed Z ed Z ed Zed Zed ZdZddZddZdZdZdZdZddZdZdZdZdZdZdZdZ RS(sRepresents Formal Power Series of a function. No computation is performed. This class should only to be used to represent a series. No checks are performed. For computing a series use :func:`fps`. See Also ======== sympy.series.formal.fps cG stt|}tj||S(N(RRRt__new__(tclsR((s2lib/python2.7/site-packages/sympy/series/formal.pyRscC s |jdS(Ni(R(tself((s2lib/python2.7/site-packages/sympy/series/formal.pytfunctionscC s |jdS(Ni(R(R((s2lib/python2.7/site-packages/sympy/series/formal.pyR<scC s |jdS(Ni(R(R((s2lib/python2.7/site-packages/sympy/series/formal.pyRscC s |jdS(Ni(R(R((s2lib/python2.7/site-packages/sympy/series/formal.pyRscC s|jddS(Nii(R(R((s2lib/python2.7/site-packages/sympy/series/formal.pyRKscC s|jddS(Nii(R(R((s2lib/python2.7/site-packages/sympy/series/formal.pyRscC s|jddS(Nii(R(R((s2lib/python2.7/site-packages/sympy/series/formal.pyRFscC s tdtS(Ni(RR(R((s2lib/python2.7/site-packages/sympy/series/formal.pytintervalscC s |jjS(N(Rtinf(R((s2lib/python2.7/site-packages/sympy/series/formal.pyRscC s |jjS(N(Rtsup(R((s2lib/python2.7/site-packages/sympy/series/formal.pytstopscC stS(N(R(R((s2lib/python2.7/site-packages/sympy/series/formal.pytlengthscC scddlm}|j|j}}|jd}||j|j||j|jf}|j|S(s0Returns an infinite representation of the seriesi(tSumi( tsympy.concreteRRKRt variablestformulaRRRF(RRRKRR=tinf_sum((s2lib/python2.7/site-packages/sympy/series/formal.pytinfinites  (cC s?|j|jdj\}}|j|js;tjS|S(s!Returns the power of x in a term.i(R1R<R2R,RR+(RRPRIR((s2lib/python2.7/site-packages/sympy/series/formal.pyt _get_pow_xs"icC sg}x{t|D]m\}}|j|}||kr>Pq|jtkra||dkraPq|tjk r|j|qqWt|S(sTruncated series as polynomial. Returns series sexpansion of ``f`` upto order ``O(x**n)`` as a polynomial(without ``O`` term). i(RMRRRbRR+R:R(RRORBR@RCtxp((s2lib/python2.7/site-packages/sympy/series/formal.pyt polynomials cC sv|dkrt|S|j|j}}|jj|}|tjkrVtj}n|j |t |||fS(sTruncated series. Returns truncated series expansion of f upto order ``O(x**n)``. If n is ``None``, returns an infinite iterator. N( R6titerR<RRR4RtNegativeInfinityRRR(RROR<Rtpt_xk((s2lib/python2.7/site-packages/sympy/series/formal.pyttruncates   cC sy.|jj|}|jj|j}Wntk rJtj}n X||}|jrtj}xwtj |jD]c}|j |}|dkr|dkr||7}qz||krz||dkrz||7}qzqzW||7}n|j |j S(Nii( RR4RKtsimplifyt IndexErrorRR+RFRR.RReR<(RtptRtpt_akRPRFRCR((s2lib/python2.7/site-packages/sympy/series/formal.pyt _eval_terms        cC s |j}|j|r|SdS(N(R<R,(RtoldtnewR<((s2lib/python2.7/site-packages/sympy/series/formal.pyt _eval_subss cC s(x!|D]}|tjk r|SqWdS(N(RR+(RR<RC((s2lib/python2.7/site-packages/sympy/series/formal.pyt_eval_as_leading_terms c C s|jj|}|jj|}|j|jj}|j}|jd}|jj|rg}xq|jj D]c\}} t j } x8t j |D]'} |j| } | | || 7} qW|j| | fqtWt|}t|j||d||jd|jf}n9t|j|j||d||jd|jf}|j||j|j|j||j|fS(Nii(RR)RFRRRRKRR,RRR+RR.R:RR R9RRtfuncR<RR( RR<R;RFtpow_xkRKR=tformRRqRRCR((s2lib/python2.7/site-packages/sympy/series/formal.pyt_eval_derivatives$    5cK sddlm}|dkr(|j}nt|rD||j|S||j|}||j|}|||j|d7}|j|j j }|j }|j d}|j j |r~g} xu|j jD]g\} } tj} x<tj| D]+} |j| }| | ||d7} qW| j| | fqWt| } t| j||d||jd|jf}n=t|j |dj||d||jd|jf}|j||j|j|j||j |fS(sBIntegrate Formal Power Series. Examples ======== >>> from sympy import fps, sin, integrate >>> from sympy.abc import x >>> f = fps(sin(x)) >>> f.integrate(x).truncate() -1 + x**2/2 - x**4/24 + O(x**6) >>> integrate(f, (x, 0, 1)) 1 - cos(1) i(R$iiN(R'R$R6R<RRRFR8RRRRKRR,RRR+RR.R:RR R9RRRRR(RR<tkwargsR$R;RFRRKR=RRRqRRCR((s2lib/python2.7/site-packages/sympy/series/formal.pyR$2s0       5!c C st|}t|tr|j|jkr<tdn!|j|jkr]tdn|j|j}}|j|jj||}|j|j kr|S|j |j }|j j |j j kr|j }|j j |j j }}n"|j }|j j |j j }}t gt |d||!|j||!D]} | d| d^q9} |j|j| } |j|||j|j||j| fS|j|js|j|}|j|} |j||j|j|j|j |j| fSt ||S(Ns9Both series should be calculated from the same direction.s6Both series should be calculated about the same point.ii(RR0RRRRR<RR9RRKRRRaRRFRR,( RtotherR<tyR;RKRRZRRRRF((s2lib/python2.7/site-packages/sympy/series/formal.pyt__add__^s2   I+  cC s |j|S(N(R(RR((s2lib/python2.7/site-packages/sympy/series/formal.pyt__radd__scC s:|j|j |j|j|j|j |j|j fS(N(RRR<RRRKRRF(R((s2lib/python2.7/site-packages/sympy/series/formal.pyt__neg__scC s|j| S(N(R(RR((s2lib/python2.7/site-packages/sympy/series/formal.pyt__sub__scC s| j|S(N(R(RR((s2lib/python2.7/site-packages/sympy/series/formal.pyt__rsub__scC st|}|j|jr+t||S|j|}|jj|}|j|}|j||j|j |j ||j |fS(N( RR,R<R RRKt coeff_mulRFRRRR(RRR;RKRF((s2lib/python2.7/site-packages/sympy/series/formal.pyt__mul__s    cC s |j|S(N(R(RR((s2lib/python2.7/site-packages/sympy/series/formal.pyt__rmul__sN(!t__name__t __module__t__doc__RtpropertyRR<RRRKRRFRRRRRRRRRRRRR6R$RRRRRRR(((s2lib/python2.7/site-packages/sympy/series/formal.pyRs:          , &     c C st|}|dkr[|j}t|dkrB|j}q[|sL|Stdnt||||||||} | dkr|St||||| S(sCGenerates Formal Power Series of f. Returns the formal series expansion of ``f`` around ``x = x0`` with respect to ``x`` in the form of a ``FormalPowerSeries`` object. Formal Power Series is represented using an explicit formula computed using different algorithms. See :func:`compute_fps` for the more details regarding the computation of formula. Parameters ========== x : Symbol, optional If x is None and ``f`` is univariate, the univariate symbols will be supplied, otherwise an error will be raised. x0 : number, optional Point to perform series expansion about. Default is 0. dir : {1, -1, '+', '-'}, optional If dir is 1 or '+' the series is calculated from the right and for -1 or '-' the series is calculated from the left. For smooth functions this flag will not alter the results. Default is 1. hyper : {True, False}, optional Set hyper to False to skip the hypergeometric algorithm. By default it is set to False. order : int, optional Order of the derivative of ``f``, Default is 4. rational : {True, False}, optional Set rational to False to skip rational algorithm. By default it is set to True. full : {True, False}, optional Set full to True to increase the range of rational algorithm. See :func:`rational_algorithm` for details. By default it is set to False. Examples ======== >>> from sympy import fps, O, ln, atan >>> from sympy.abc import x Rational Functions >>> fps(ln(1 + x)).truncate() x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6) >>> fps(atan(x), full=True).truncate() x - x**3/3 + x**5/5 + O(x**6) See Also ======== sympy.series.formal.FormalPowerSeries sympy.series.formal.compute_fps is multivariate formal power seriesN(RR6RR_R7tNotImplementedErrorRR( R;R<RRRR>RR%tfreeR((s2lib/python2.7/site-packages/sympy/series/formal.pytfpss9   ! 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