a ам>> 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) cSs6|j}t|Гdkr|абS|s&tdГStd|ГВdS)Nr┌kzм 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)))┌free_symbols┌len┌popr ┌ ValueError)r┌freer$r$r%┌_find_x>s¤ z _process_limits.._find_x)NNNNщrzInvalid limits given: %sz.Both the start and end value should be bounded)rrrr*┌ isinstancer r,┌strr┌NegativeInfinity┌Infinityr)rrr.r!┌start┌stopZ unboundedr$r$r%┌_process_limits's r6c sоddД}ЗЗfddД}ttt|ГГГ}|аб}tdddДddДgd НЙtd ЗfddДgd НЙ|dD]@}|абd}|D]*} || ИГsx|| И|Гsxd |fSqxqdd|fS)NcSs ||jvSйNйr))┌exprsr!r$r$r%┌check_fxaszfinite_check..check_fxcsBt|ttfГr>|jd}|аИt||Ибdur:dSdSdS)NrTF)r0rr ┌args┌matchr)┌_exprr!r"Zsincos_args)┌a┌br$r%┌check_sincosds z"finite_check..check_sincosr>cSs|jSr7йZ is_Integerйr(r$r$r%┌pєzfinite_check..cSs |tjkSr7йrrrBr$r$r%rCprDйZ propertiesr?cs И|jvSr7r8rBйr!r$r%rCqrDrFT)rrr┌as_coeff_addrZas_coeff_mul) ┌fr!r"r:r@r=Z add_coeff┌sZ mul_coeffs┌tr$)r>r?r!r%┌finite_check_s rLc@seZdZdZddДZeddДГZeddДГZedd ДГZed dДГZ edd ДГZ eddДГZeddДГZeddДГZ eddДГZeddДГZeddДГZddДZd5ddДZd6dd ДZd!d"ДZd#d$ДZd%d&ДZd'd(ДZd7d+d,ДZd-d.ДZd/d0ДZd1d2ДZd3d4ДZd)S)8┌ FourierSeriesa9Represents Fourier sine/cosine series. Explanation =========== 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 cGstt|Г}tj|g|вRОSr7)┌maprr┌__new__)┌clsr;r$r$r%rOНs zFourierSeries.__new__cCs |jdSйNrйr;й┌selfr$r$r%┌functionСszFourierSeries.functioncCs|jddSйNrrrRrSr$r$r%r!ХszFourierSeries.xcCs|jdd|jddfS)NrrrRrSr$r$r%┌periodЩszFourierSeries.periodcCs|jddS)NrrrRrSr$r$r%r#ЭszFourierSeries.a0cCs|jddS)NrrrRrSr$r$r%┌anбszFourierSeries.ancCs|jddS)NrrRrSr$r$r%┌bnеszFourierSeries.bncCs tdtГSrQ)rrrSr$r$r%┌intervalйszFourierSeries.intervalcCs|jjSr7)rZ┌infrSr$r$r%r4нszFourierSeries.startcCs|jjSr7)rZ┌suprSr$r$r%r5▒szFourierSeries.stopcCstSr7)rrSr$r$r%┌length╡szFourierSeries.lengthcCst|jd|jdГdS)Nrrr)┌absrWrSr$r$r%r"╣szFourierSeries.LcCs|j}|а|бr|SdSr7)r!Zhas)rT┌old┌newr!r$r$r%┌ _eval_subs╜s zFourierSeries._eval_subsr/cCsJ|durt|ГSg}|D](}t|Г|kr,qB|tjur|а|бqt|ОS)a═ 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)┌iterr*rr┌appendr)rTr ┌termsrKr$r$r%┌truncate┬s zFourierSeries.truncatecs&ЗfddДt|dИЕГDГ}t|ОS)a Return :math:`\sigma`-approximation of Fourier series with respect to order n. Explanation =========== 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 cs.g|]&\}}|tjurtt|ИГ|СqSr$)rrrr)┌.0┌irKйr r$r%┌ 0s z5FourierSeries.sigma_approximation..N)┌ enumerater)rTr rdr$rhr%┌sigma_approximationюsBz!FourierSeries.sigma_approximationcCs\t|Г|j}}||jvr*td||fГВ|j|}|j|}|а||jd||j|j fбS)a╩ Shift the function by a term independent of x. Explanation =========== f(x) -> f(x) + s This is fast, if Fourier series of f(x) is already computed. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.shift(1).truncate() -4*cos(x) + cos(2*x) + 1 + pi**2/3 · '%s' should be independent of %sr) rr!r)r,r#rUrr;rXrY)rTrJr!r#┌sfuncr$r$r%┌shift4s zFourierSeries.shiftcCs|t|Г|j}}||jvr*td||fГВ|jа|||б}|jа|||б}|jа|||б}|а||j d|j ||fбS)a─ Shift x by a term independent of x. Explanation =========== f(x) -> f(x + s) This is fast, if Fourier series of f(x) is already computed. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.shiftx(1).truncate() -4*cos(x + 1) + cos(2*x + 2) + pi**2/3 rlrйrr!r)r,rXrrYrUrr;r#йrTrJr!rXrYrmr$r$r%┌shiftxSs zFourierSeries.shiftxcCstt|Г|j}}||jvr*td||fГВ|jа|б}|jа|б}|j|}|jd|}|а ||jd|||fбS)a╩ Scale the function by a term independent of x. Explanation =========== f(x) -> s * f(x) This is fast, if Fourier series of f(x) is already computed. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.scale(2).truncate() -8*cos(x) + 2*cos(2*x) + 2*pi**2/3 rlrr) rr!r)r,rXZ coeff_mulrYr#r;r)rTrJr!rXrYr#rmr$r$r%┌scaless zFourierSeries.scalecCs|t|Г|j}}||jvr*td||fГВ|jа|||б}|jа|||б}|jа|||б}|а||j d|j ||fбS)a╝ Scale x by a term independent of x. Explanation =========== f(x) -> f(s*x) This is fast, if Fourier series of f(x) is already computed. Examples ======== >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.scalex(2).truncate() -4*cos(2*x) + cos(4*x) + pi**2/3 rlrrorpr$r$r%┌scalexФs zFourierSeries.scalexNrcCs |D]}|tjur|SqdSr7rE)rTr!Zlogx┌cdirrKr$r$r%┌_eval_as_leading_term┤s z#FourierSeries._eval_as_leading_termcCs&|dkr|jS|jа|б|jа|бSrQ)r#rXZcoeffrY)rT┌ptr$r$r%┌ _eval_term╣szFourierSeries._eval_termcCs |аdбS)Nщ )rrrSr$r$r%┌__neg__╛szFourierSeries.__neg__cCsШt|tГrО|j|jkrtdГВ|j|j}}|j|jа||б}|j|jvrP|S|j|j}|j |j }|j |j }|а||jd|||fбSt ||ГS)N·(Both the series should have same periodsr)r0rMrWr,r!rUrr)rXrYr#rr;r)rT┌otherr!┌yrUrXrYr#r$r$r%┌__add__┴s zFourierSeries.__add__cCs|а|бSr7)r})rTr{r$r$r%┌__sub__╘szFourierSeries.__sub__)r/)r/)Nr)┌__name__┌ __module__┌__qualname__┌__doc__rO┌propertyrUr!rWr#rXrYrZr4r5r]r"rarerkrnrqrrrsrurwryr}r~r$r$r$r%rM|sH , F ! rMc@sXeZdZdZddДZeddДГZeddДГZdd ДZd dДZ dd ДZ ddДZddДZdS)┌FiniteFourierSeriesa∙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 csДt|Г}t|Г}t|Г}t|tГr0t|ГdkРst|аб\}}|tddД|DГО}|jdddddНаб\}}|dЙt|d|dГd} td d dДddДgd Н} tdЗfddДgd Н}t Г}t Г} |D]Ь}|а |t| t| ИГб}|а |t | t| ИГб}|Рr4|||а|| tjб||| <q╩|Рr^||| а|| tjб| || <q╩||7}q╩t||| Г}tа||||бS)Nr/cSsg|]}t|ГСqSr$)r)rfrgr$r$r%rirDz/FiniteFourierSeries.__new__..F)ZtrigZ power_baseZ power_exp┌logrrrr>cSs|jSr7rArBr$r$r%rC rDz-FiniteFourierSeries.__new__..cSs |tjuSr7rErBr$r$r%rC rDrFr?cs И|jvSr7r8rBrGr$r%rCrD)rr0rr*rHr┌expandr^r┌dictr<r rr┌getrrrrO)rPrIrr9┌c┌eZrexprr#Zexp_lsr"r>r?rXrY┌prK┌qr$rGr%rO s.$$ zFiniteFourierSeries.__new__cCsB|jr dnd}|tt|jабГаt|jабГбГd7}td|ГSrV)r#┌max┌setrX┌keys┌unionrYr)rT┌_lengthr$r$r%rZ"s*zFiniteFourierSeries.intervalcCs|j|jSr7)r5r4rSr$r$r%r](szFiniteFourierSeries.lengthcCsdt|Г|j}}||jvr*td||fГВ|аба|||б}|jа|||б}|а||jd|бSйNrlrй rr!r)r,rerrUrr;йrTrJr!r=rmr$r$r%rq,s zFiniteFourierSeries.shiftxcCsTt|Г|j}}||jvr*td||fГВ|аб|}|j|}|а||jd|бSrТ)rr!r)r,rerUrr;rФr$r$r%rr7s zFiniteFourierSeries.scalecCsdt|Г|j}}||jvr*td||fГВ|аба|||б}|jа|||б}|а||jd|бSrТrУrФr$r$r%rsBs zFiniteFourierSeries.scalexcCsb|dkr|jS|jа|tjбt|t|j|jГ|j а|tjбt |t|j|jГ}|SrQ)r#rXrИrrr rr"r!rYr)rTrvZ_termr$r$r%rwMs&& zFiniteFourierSeries._eval_termcCsМt|tГr&|аt|j|jdddНбSt|tГrИ|j|jkrDtdГВ|j |j }}|j|jа ||б}|j |jvrv|St||jddНSdS)NrF)┌finiterz)r)r0rMr}┌fourier_seriesrUr;rДrWr,r!rr))rTr{r!r|rUr$r$r%r}Us zFiniteFourierSeries.__add__N) rrАrБrВrOrГrZr]rqrrrsrwr}r$r$r$r%rД╪s&# rДNTc Cs:t|Г}t||Г}|d}||jvr(|S|rdt|d|dГd}t|||Г\}}|rdt|||ГStdГ}|d|dd}|jРr|а||б} || kr╬t |||Г\} }t ddtfГ}t||| ||fГS|| kРrt j} t ddtfГ}t|||Г}t||| ||fГSt |||Г\} }t|||Г}t||| ||fГS)a_Computes the Fourier trigonometric series expansion. Explanation =========== Fourier trigonometric series of $f(x)$ over the interval $(a, b)$ is defined as: .. math:: \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(\frac{2n \pi x}{L}) + b_n \sin(\frac{2n \pi x}{L})) where the coefficients are: .. math:: L = b - a .. math:: a_0 = \frac{2}{L} \int_{a}^{b}{f(x) dx} .. math:: a_n = \frac{2}{L} \int_{a}^{b}{f(x) \cos(\frac{2n \pi x}{L}) dx} .. math:: b_n = \frac{2}{L} \int_{a}^{b}{f(x) \sin(\frac{2n \pi x}{L}) dx} The condition whether the function $f(x)$ given should be periodic or not is more than necessary, because it is sufficient to consider the series to be converging to $f(x)$ only in the given interval, not throughout the whole real line. This also brings a lot of ease for the computation because you don't have to make $f(x)$ artificially periodic by wrapping it with piecewise, modulo operations, but you can shape the function to look like the desired periodic function only in the interval $(a, b)$, and the computed series will automatically become the series of the periodic version of $f(x)$. This property is illustrated in the examples section below. Parameters ========== limits : (sym, start, end), optional *sym* denotes the symbol the series is computed with respect to. *start* and *end* denotes the start and the end of the interval where the fourier series converges to the given function. Default range is specified as $-\pi$ and $\pi$. Returns ======= FourierSeries A symbolic object representing the Fourier trigonometric series. Examples ======== Computing the Fourier series of $f(x) = x^2$: >>> from sympy import fourier_series, pi >>> from sympy.abc import x >>> f = x**2 >>> s = fourier_series(f, (x, -pi, pi)) >>> s1 = s.truncate(n=3) >>> s1 -4*cos(x) + cos(2*x) + pi**2/3 Shifting of the Fourier series: >>> 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 of the Fourier series: >>> 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 Computing the Fourier series of $f(x) = x$: This illustrates how truncating to the higher order gives better convergence. .. plot:: :context: reset :format: doctest :include-source: True >>> from sympy import fourier_series, pi, plot >>> from sympy.abc import x >>> f = x >>> s = fourier_series(f, (x, -pi, pi)) >>> s1 = s.truncate(n = 3) >>> s2 = s.truncate(n = 5) >>> s3 = s.truncate(n = 7) >>> p = plot(f, s1, s2, s3, (x, -pi, pi), show=False, legend=True) >>> p[0].line_color = (0, 0, 0) >>> p[0].label = 'x' >>> p[1].line_color = (0.7, 0.7, 0.7) >>> p[1].label = 'n=3' >>> p[2].line_color = (0.5, 0.5, 0.5) >>> p[2].label = 'n=5' >>> p[3].line_color = (0.3, 0.3, 0.3) >>> p[3].label = 'n=7' >>> p.show() This illustrates how the series converges to different sawtooth waves if the different ranges are specified. .. plot:: :context: close-figs :format: doctest :include-source: True >>> s1 = fourier_series(x, (x, -1, 1)).truncate(10) >>> s2 = fourier_series(x, (x, -pi, pi)).truncate(10) >>> s3 = fourier_series(x, (x, 0, 1)).truncate(10) >>> p = plot(x, s1, s2, s3, (x, -5, 5), show=False, legend=True) >>> p[0].line_color = (0, 0, 0) >>> p[0].label = 'x' >>> p[1].line_color = (0.7, 0.7, 0.7) >>> p[1].label = '[-1, 1]' >>> p[2].line_color = (0.5, 0.5, 0.5) >>> p[2].label = '[-pi, pi]' >>> p[3].line_color = (0.3, 0.3, 0.3) >>> p[3].label = '[0, 1]' >>> p.show() 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] https://mathworld.wolfram.com/FourierSeries.html rrrr )rr6r)r^rLrДr ┌is_zerorr&rrrMrrr') rIrrХr!r"┌ is_finiteZres_fr ┌centerZneg_fr#rXrYr$r$r%rЦfs6% rЦ)NT)*rВZsympy.core.numbersrrZsympy.core.symbolrZsympy.core.exprrZsympy.core.addrZsympy.core.containersrZsympy.core.singletonrr r Zsympy.core.sympifyrZ(sympy.functions.elementary.trigonometricrr rZsympy.series.series_classrZsympy.series.sequencesrZsympy.sets.setsrZsympy.simplify.furrrrZsympy.utilities.iterablesrr&r'r6rLrMrДrЦr$r$r$r%┌s0 8^