ó ~9­\c@ s‹dZddlmZmZddlmZmZmZmZddl m Z ddl m Z ddl mZddlmZddlmZdd lmZmZdd lmZdd lmZmZmZdd lmZdd lmZddl m!Z!ddl"m#Z#m$Z$m%Z%m&Z&m'Z'd„Z(d„Z)d„Z*d„Z+defd„ƒYZ,de,fd„ƒYZ-e.e/d„Z0dS(sFourier Seriesiÿÿÿÿ(tprint_functiontdivision(tpitootWildtBasic(tExpr(tAdd(t is_sequence(tTuple(tS(tDummytSymbol(tsympify(tsintcostsinc(t SeriesBase(t SeqFormula(tInterval(tTR8tTR2tTR1tTR10t sincos_to_sumc C s¶ddlm}|d|d|d}}td|t||ƒ}d|||||ƒ|}|j|tjƒd}|td|||||ƒ||dtfƒfS(s,Returns the cos sequence in a Fourier seriesiÿÿÿÿ(t integrateiii( tsympy.integralsRRRtsubsR tZeroRR( tfunctlimitstnRtxtLtcos_termtformulata0((s3lib/python2.7/site-packages/sympy/series/fourier.pytfourier_cos_seqscC sxddlm}|d|d|d}}td|t||ƒ}td|||||ƒ||dtfƒS(s,Returns the sin sequence in a Fourier seriesiÿÿÿÿ(Riii(RRRRRR(RRRRR R!tsin_term((s3lib/python2.7/site-packages/sympy/series/fourier.pytfourier_sin_seqs cC s2d„}d\}}}|dkrB||ƒt t}}}nt|tƒr¥t|ƒdkru|\}}}q¥t|ƒdkr¥||ƒ}|\}}q¥nt|tƒ sÍ|dksÍ|dkrætdt|ƒƒ‚nt j t j g}||ks||krtdƒ‚nt |||fƒS(s Limits should be of the form (x, start, stop). x should be a symbol. Both start and stop should be bounded. * If x is not given, x is determined from func. * If limits is None. Limit of the form (x, -pi, pi) is returned. Examples ======== >>> from sympy import pi >>> from sympy.series.fourier import _process_limits as pari >>> from sympy.abc import x >>> pari(x**2, (x, -2, 2)) (x, -2, 2) >>> pari(x**2, (-2, 2)) (x, -2, 2) >>> pari(x**2, None) (x, -pi, pi) cS sI|j}t|ƒdkr%|jƒS|s5tdƒStd|ƒ‚dS(Nitks¬ specify dummy variables for %s. If the function contains more than one free symbol, a dummy variable should be supplied explicitly e.g. FourierSeries(m*n**2, (n, -pi, pi))(t free_symbolstlentpopR t ValueError(Rtfree((s3lib/python2.7/site-packages/sympy/series/fourier.pyt_find_x=s   iisInvalid limits given: %ss.Both the start and end value should be boundedN(NNN( tNoneRRR R*t isinstanceR R,tstrR tNegativeInfinitytInfinityR (RRR.R tstarttstopt unbounded((s3lib/python2.7/site-packages/sympy/series/fourier.pyt_process_limits(s    (c  sæd„}‡‡fd†}ttt|ƒƒƒ}|jƒ}tddd„d„gƒ‰tdd‡fd†gƒ‰x^|d D]R}|jƒd }x9|D]1} || ˆƒpÇ|| ˆ|ƒs£t|fSq£Wq†Wt|fS( NcS s ||jkS(N(R)(texprsR ((s3lib/python2.7/site-packages/sympy/series/fourier.pytcheck_fx`sc sVt|ttfƒrR|jd}|jˆt||ˆƒdk rKtStSndS(Ni( R0RRtargstmatchRR/tTruetFalse(t_exprR R!t sincos_args(tatb(s3lib/python2.7/site-packages/sympy/series/fourier.pyt check_sincoscs  %R@t propertiescS s|jS(N(t is_Integer(R(((s3lib/python2.7/site-packages/sympy/series/fourier.pytotcS s |tjkS(N(R R(R(((s3lib/python2.7/site-packages/sympy/series/fourier.pyREoRFRAc s ˆ|jkS(N(R)(R((R (s3lib/python2.7/site-packages/sympy/series/fourier.pyREpRFi(RRRt as_coeff_addRt as_coeff_mulR=R<( tfR R!R9RBR>t add_coefftst mul_coeffstt((R@RAR s3lib/python2.7/site-packages/sympy/series/fourier.pyt finite_check^s    !t FourierSeriescB s.eZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ ed„ƒZ ed„ƒZ ed „ƒZ ed „ƒZ ed „ƒZed „ƒZd „Zdd„Zdd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„ZRS(sRepresents Fourier sine/cosine series. This class only represents a fourier series. No computation is performed. For how to compute Fourier series, see the :func:`fourier_series` docstring. See Also ======== sympy.series.fourier.fourier_series cG stt|ƒ}tj||ŒS(N(tmapR Rt__new__(tclsR:((s3lib/python2.7/site-packages/sympy/series/fourier.pyRQ‰scC s |jdS(Ni(R:(tself((s3lib/python2.7/site-packages/sympy/series/fourier.pytfunctionscC s|jddS(Nii(R:(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pyR ‘scC s |jdd|jddfS(Nii(R:(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pytperiod•scC s|jddS(Nii(R:(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pyR$™scC s|jddS(Nii(R:(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pytanscC s|jddS(Ni(R:(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pytbn¡scC s tdtƒS(Ni(RR(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pytinterval¥scC s |jjS(N(RXtinf(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pyR4©scC s |jjS(N(RXtsup(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pyR5­scC stS(N(R(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pytlength±scC s t|jd|jdƒdS(Niii(tabsRU(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pyR!µscC s |j}|j|ƒr|SdS(N(R thas(RStoldtnewR ((s3lib/python2.7/site-packages/sympy/series/fourier.pyt _eval_subs¹s icC sl|dkrt|ƒSg}xC|D];}t|ƒ|kr?Pn|tjk r#|j|ƒq#q#Wt|ŒS(sÅ Return the first n nonzero terms of the series. If n is None return an iterator. Parameters ========== n : int or None Amount of non-zero terms in approximation or None. Returns ======= Expr or iterator Approximation of function expanded into Fourier series. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x, (x, -pi, pi)) >>> s.truncate(4) 2*sin(x) - sin(2*x) + 2*sin(3*x)/3 - sin(4*x)/2 See Also ======== sympy.series.fourier.FourierSeries.sigma_approximation N(R/titerR*R RtappendR(RSRttermsRM((s3lib/python2.7/site-packages/sympy/series/fourier.pyttruncate¾s   cC sTgt|| ƒD]3\}}|tjk rtt||ƒ|^q}t|ŒS(sð Return :math:`\sigma`-approximation of Fourier series with respect to order n. Sigma approximation adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at discontinuities. A sigma-approximated summation for a Fourier series of a T-periodical function can be written as .. math:: s(\theta) = \frac{1}{2} a_0 + \sum _{k=1}^{m-1} \operatorname{sinc} \Bigl( \frac{k}{m} \Bigr) \cdot \left[ a_k \cos \Bigl( \frac{2\pi k}{T} \theta \Bigr) + b_k \sin \Bigl( \frac{2\pi k}{T} \theta \Bigr) \right], where :math:`a_0, a_k, b_k, k=1,\ldots,{m-1}` are standard Fourier series coefficients and :math:`\operatorname{sinc} \Bigl( \frac{k}{m} \Bigr)` is a Lanczos :math:`\sigma` factor (expressed in terms of normalized :math:`\operatorname{sinc}` function). Parameters ========== n : int Highest order of the terms taken into account in approximation. Returns ======= Expr Sigma approximation of function expanded into Fourier series. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x, (x, -pi, pi)) >>> s.sigma_approximation(4) 2*sin(x)*sinc(pi/4) - 2*sin(2*x)/pi + 2*sin(3*x)*sinc(3*pi/4)/3 See Also ======== sympy.series.fourier.FourierSeries.truncate Notes ===== The behaviour of :meth:`~sympy.series.fourier.FourierSeries.sigma_approximation` is different from :meth:`~sympy.series.fourier.FourierSeries.truncate` - it takes all nonzero terms of degree smaller than n, rather than first n nonzero ones. References ========== .. [1] https://en.wikipedia.org/wiki/Gibbs_phenomenon .. [2] https://en.wikipedia.org/wiki/Sigma_approximation (t enumerateR RRRR(RSRtiRMRc((s3lib/python2.7/site-packages/sympy/series/fourier.pytsigma_approximationès=-cC st|ƒ|j}}||jkr>td||fƒ‚n|j|}|j|}|j||jd||j|j fƒS(s˜Shift the function by a term independent of x. f(x) -> f(x) + s This is fast, if Fourier series of f(x) is already computed. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.shift(1).truncate() -4*cos(x) + cos(2*x) + 1 + pi**2/3 s '%s' should be independent of %si( R R R)R,R$RTRR:RVRW(RSRKR R$tsfunc((s3lib/python2.7/site-packages/sympy/series/fourier.pytshift)s   cC s¯t|ƒ|j}}||jkr>td||fƒ‚n|jj|||ƒ}|jj|||ƒ}|jj|||ƒ}|j||j d|j ||fƒS(s’Shift x by a term independent of x. f(x) -> f(x + s) This is fast, if Fourier series of f(x) is already computed. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.shiftx(1).truncate() -4*cos(x + 1) + cos(2*x + 2) + pi**2/3 s '%s' should be independent of %si( R R R)R,RVRRWRTRR:R$(RSRKR RVRWRh((s3lib/python2.7/site-packages/sympy/series/fourier.pytshiftxDscC s£t|ƒ|j}}||jkr>td||fƒ‚n|jj|ƒ}|jj|ƒ}|j|}|jd|}|j ||jd|||fƒS(s˜Scale the function by a term independent of x. f(x) -> s * f(x) This is fast, if Fourier series of f(x) is already computed. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.scale(2).truncate() -8*cos(x) + 2*cos(2*x) + 2*pi**2/3 s '%s' should be independent of %sii( R R R)R,RVt coeff_mulRWR$R:R(RSRKR RVRWR$Rh((s3lib/python2.7/site-packages/sympy/series/fourier.pytscale`s cC s¯t|ƒ|j}}||jkr>td||fƒ‚n|jj|||ƒ}|jj|||ƒ}|jj|||ƒ}|j||j d|j ||fƒS(sŠScale x by a term independent of x. f(x) -> f(s*x) This is fast, if Fourier series of f(x) is already computed. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.scalex(2).truncate() -4*cos(2*x) + cos(4*x) + pi**2/3 s '%s' should be independent of %si( R R R)R,RVRRWRTRR:R$(RSRKR RVRWRh((s3lib/python2.7/site-packages/sympy/series/fourier.pytscalex}scC s(x!|D]}|tjk r|SqWdS(N(R R(RSR RM((s3lib/python2.7/site-packages/sympy/series/fourier.pyt_eval_as_leading_term™s cC s3|dkr|jS|jj|ƒ|jj|ƒS(Ni(R$RVtcoeffRW(RStpt((s3lib/python2.7/site-packages/sympy/series/fourier.pyt _eval_termžs cC s |jdƒS(Niÿÿÿÿ(Rl(RS((s3lib/python2.7/site-packages/sympy/series/fourier.pyt__neg__£scC sÕt|tƒrÈ|j|jkr0tdƒ‚n|j|j}}|j|jj||ƒ}|j|jkru|S|j|j}|j |j }|j |j }|j ||j d|||fƒSt ||ƒS(Ns(Both the series should have same periodsi(R0RORUR,R RTRR)RVRWR$RR:R(RStotherR tyRTRVRWR$((s3lib/python2.7/site-packages/sympy/series/fourier.pyt__add__¦s#cC s|j| ƒS(N(Ru(RSRs((s3lib/python2.7/site-packages/sympy/series/fourier.pyt__sub__¹s(t__name__t __module__t__doc__RQtpropertyRTR RUR$RVRWRXR4R5R[R!R`RdRgRiRjRlRmRnRqRrRuRv(((s3lib/python2.7/site-packages/sympy/series/fourier.pyRO{s2    * A        tFiniteFourierSeriescB sbeZdZd„Zed„ƒZed„ƒZd„Zd„Zd„Z d„Z d„Z RS( s÷Represents Finite Fourier sine/cosine series. For how to compute Fourier series, see the :func:`fourier_series` docstring. Parameters ========== f : Expr Expression for finding fourier_series limits : ( x, start, stop) x is the independent variable for the expression f (start, stop) is the period of the fourier series exprs: (a0, an, bn) or Expr a0 is the constant term a0 of the fourier series an is a dictionary of coefficients of cos terms an[k] = coefficient of cos(pi*(k/L)*x) bn is a dictionary of coefficients of sin terms bn[k] = coefficient of sin(pi*(k/L)*x) or exprs can be an expression to be converted to fourier form Methods ======= This class is an extension of FourierSeries class. Please refer to sympy.series.fourier.FourierSeries for further information. See Also ======== sympy.series.fourier.FourierSeries sympy.series.fourier.fourier_series c  st|ƒtko!t|ƒdksÞ|jƒ\}}|tg|D]}t|ƒ^qCŒ}|jdtdtdtdtƒjƒ\}} |d‰t|d|dƒd} t d d d „d „gƒ} t d d ‡fd†gƒ} t ƒ} t ƒ}xÆ| D]¾}|j | t | t | ˆƒƒ}|j | t| t | ˆƒƒ}|rŒ|| | j|| tjƒ| || td||fƒ‚n|jƒj|||ƒ}|jj|||ƒ}|j||jd|ƒS(Ns '%s' should be independent of %si( R R R)R,RdRRTRR:(RSRKR R>Rh((s3lib/python2.7/site-packages/sympy/series/fourier.pyRj s cC sut|ƒ|j}}||jkr>td||fƒ‚n|jƒ|}|j|}|j||jd|ƒS(Ns '%s' should be independent of %si(R R R)R,RdRTRR:(RSRKR R>Rh((s3lib/python2.7/site-packages/sympy/series/fourier.pyRls  cC st|ƒ|j}}||jkr>td||fƒ‚n|jƒj|||ƒ}|jj|||ƒ}|j||jd|ƒS(Ns '%s' should be independent of %si( R R R)R,RdRRTRR:(RSRKR R>Rh((s3lib/python2.7/site-packages/sympy/series/fourier.pyRm#s cC s}|dkr|jS|jj|tjƒt|t|j|jƒ|j j|tjƒt |t|j|jƒ}|S(Ni( R$RVR„R RRRR!R RWR(RSRpt_term((s3lib/python2.7/site-packages/sympy/series/fourier.pyRq.s  15cC sÅt|tƒr5|jt|j|jddtƒƒSt|tƒrÁ|j|jkret dƒ‚n|j |j }}|j|jj ||ƒ}|j |j krª|St|d|jdƒSdS(Nitfinites(Both the series should have same periodsR( R0RORutfourier_seriesRTR:R=R{RUR,R RR)(RSRsR RtRT((s3lib/python2.7/site-packages/sympy/series/fourier.pyRu6s ( RwRxRyRQRzRXR[RjRlRmRqRu(((s3lib/python2.7/site-packages/sympy/series/fourier.pyR{½s# ! c C sit|ƒ}t||ƒ}|d}||jkr8|S|r‹t|d|dƒd}t|||ƒ\}}|r‹t|||ƒSntdƒ}|j|| ƒ}||kræt|||ƒ\} } t ddt fƒ} nj|| kr&t j } t ddt fƒ} t |||ƒ} n*t|||ƒ\} } t |||ƒ} t||| | | fƒS(s©Computes Fourier sine/cosine series expansion. Returns a :class:`FourierSeries` object. Examples ======== >>> from sympy import fourier_series, pi, cos >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.truncate(n=3) -4*cos(x) + cos(2*x) + pi**2/3 Shifting >>> s.shift(1).truncate() -4*cos(x) + cos(2*x) + 1 + pi**2/3 >>> s.shiftx(1).truncate() -4*cos(x + 1) + cos(2*x + 2) + pi**2/3 Scaling >>> s.scale(2).truncate() -8*cos(x) + 2*cos(2*x) + 2*pi**2/3 >>> s.scalex(2).truncate() -4*cos(2*x) + cos(4*x) + pi**2/3 Notes ===== Computing Fourier series can be slow due to the integration required in computing an, bn. It is faster to compute Fourier series of a function by using shifting and scaling on an already computed Fourier series rather than computing again. e.g. If the Fourier series of ``x**2`` is known the Fourier series of ``x**2 - 1`` can be found by shifting by ``-1``. See Also ======== sympy.series.fourier.FourierSeries References ========== .. [1] mathworld.wolfram.com/FourierSeries.html iiiR(R R7R)R\RNR{R RR%RRR RR'RO( RIRR‘R R!t is_finitetres_fRtneg_fR$RVRW((s3lib/python2.7/site-packages/sympy/series/fourier.pyR’Gs,6      N(1Ryt __future__RRtsympyRRRRtsympy.core.exprRtsympy.core.addRtsympy.core.compatibilityRtsympy.core.containersR tsympy.core.singletonR tsympy.core.symbolR R tsympy.core.sympifyR t(sympy.functions.elementary.trigonometricRRRtsympy.series.series_classRtsympy.series.sequencesRtsympy.sets.setsRtsympy.simplify.fuRRRRRR%R'R7RNROR{R/R<R’(((s3lib/python2.7/site-packages/sympy/series/fourier.pyts,"( 6 ÿCŠ