o *ήceY@sdZddlmZmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZmZdd lmZdd lmZmZmZdd lmZdd lmZdd lmZddlmZddZddZddZ ddZ!GdddeZ"Gddde"Z#dddZ$dS) zFourier Series)oopi)Wild)Expr)Add)Tuple)S)DummySymbol)sympify)sincossinc) SeriesBase) SeqFormula)Interval) is_sequencec Csddlm}|d|d|d}}td|t||}d||||||}||tjd}|td|||||||dtffS)z,Returns the cos sequence in a Fourier seriesr integrate) sympy.integralsrr rsubsrZerorr) funclimitsnrxLcos_termformulaa0r";/tmp/pip-target-vg8gfxp4/lib/python/sympy/series/fourier.pyfourier_cos_seqs r$cCsdddlm}|d|d|d}}td|t||}td|||||||dtfS)z,Returns the sin sequence in a Fourier seriesrrrr)rrr rrr)rrrrrrsin_termr"r"r#fourier_sin_seqs r&cCsdd}d\}}}|dur||t t}}}t|tr7t|dkr)|\}}}nt|dkr7||}|\}}t|trD|dusD|durLtdt|tj tj g}||vsZ||vr^tdt |||fS) a Limits should be of the form (x, start, stop). x should be a symbol. Both start and stop should be bounded. Explanation =========== * 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.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) cSs2|j}t|dkr |S|stdStd|)Nrkz 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_symbolslenpopr ValueError)rfreer"r"r#_find_x=s z _process_limits.._find_x)NNNNrzInvalid limits given: %sz.Both the start and end value should be bounded) rrrr) isinstancer r+strrNegativeInfinityInfinityr )rrr-rstartstop unboundedr"r"r#_process_limits&s       r6c sdd}fdd}ddlm}m}m}||||}|} tddd d d gd td fd d gd | dD] } | d} | D]} || s[|| |s[d|fSqFq.check_fxcsBt|ttfr|jd}|t||durdSdSdS)NrTF)r/r r argsmatchr)_exprrr sincos_args)abr"r# check_sincoscs  z"finite_check..check_sincosr)TR2TR1 sincos_to_sumr@cS|jSr7 is_Integerr'r"r"r#pzfinite_check..cSs |tjkSr7rrrIr"r"r#rJp  propertiesrAc |jvSr7r8rIrr"r#rJqrMrFT)sympy.simplify.furCrDrE as_coeff_addr as_coeff_mul) frrr:rBrCrDrEr> add_coeffs mul_coeffstr")r@rArr# finite_check^s   rZc@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ eddZ eddZ eddZeddZeddZddZd5ddZd5dd Zd!d"Zd#d$Zd%d&Zd'd(Zd6d+d,Zd-d.Zd/d0Zd1d2Zd3d4Zd)S)7 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|}tj|g|RSr7)mapr r__new__)clsr<r"r"r#r]s zFourierSeries.__new__cCs |jdSNrr<selfr"r"r#function zFourierSeries.functioncC|jddSNrrr`rar"r"r#rzFourierSeries.xcCs|jdd|jddfS)Nrrr`rar"r"r#periodzFourierSeries.periodcCre)Nrrr`rar"r"r#r!rgzFourierSeries.a0cCre)Nrrr`rar"r"r#anrgzFourierSeries.ancCs|jddS)Nrr`rar"r"r#bnrgzFourierSeries.bncCs tdtSr_)rrrar"r"r#intervalrdzFourierSeries.intervalcC|jjSr7)rlinfrar"r"r#r3zFourierSeries.startcCrmr7)rlsuprar"r"r#r4rozFourierSeries.stopcCstSr7)rrar"r"r#lengthszFourierSeries.lengthcCst|jd|jddS)Nrrr)absrhrar"r"r#rrizFourierSeries.LcCs|j}||r |SdSr7)rhas)rboldnewrr"r"r# _eval_subss zFourierSeries._eval_subsr.cCsP|durt|Sg}|D]}t||krt|S|tjur#||q t|S)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)rrappendr)rbrtermsrYr"r"r#truncates   zFourierSeries.truncatecs&fddt|dD}t|S)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||qSr")rrrr).0irYrr"r# 0s z5FourierSeries.sigma_approximation..N) enumerater)rbrryr"r}r#sigma_approximationsBz!FourierSeries.sigma_approximationcCs\t||j}}||jvrtd||f|j|}|j|}|||jd||j|j fS)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) r rr(r+r!rcrr<rjrk)rbrWrr!sfuncr"r"r#shift4s    zFourierSeries.shiftcCs|t||j}}||jvrtd||f|j|||}|j|||}|j|||}|||j d|j ||fS)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 rr r rr(r+rjrrkrcrr<r!rbrWrrjrkrr"r"r#shiftxS zFourierSeries.shiftxcCstt||j}}||jvrtd||f|j|}|j|}|j|}|jd|}| ||jd|||fS)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 rrr) r rr(r+rj coeff_mulrkr!r<r)rbrWrrjrkr!rr"r"r#scaless    zFourierSeries.scalecCs|t||j}}||jvrtd||f|j|||}|j|||}|j|||}|||j d|j ||fS)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 rrrrr"r"r#scalexrzFourierSeries.scalexNrcCs |D] }|tjur |SqdSr7rL)rbrlogxcdirrYr"r"r#_eval_as_leading_terms  z#FourierSeries._eval_as_leading_termcCs&|dkr|jS|j||j|Sr_)r!rjcoeffrk)rbptr"r"r# _eval_termszFourierSeries._eval_termcCs |dS)N)rrar"r"r#__neg__r;zFourierSeries.__neg__cCst|trG|j|jkrtd|j|j}}|j|j||}|j|jvr(|S|j|j}|j |j }|j |j }| ||j d|||fSt ||S)N(Both the series should have same periodsr)r/r[rhr+rrcrr(rjrkr!rr<r)rbotherryrcrjrkr!r"r"r#__add__s       zFourierSeries.__add__cCs || Sr7)r)rbrr"r"r#__sub__s zFourierSeries.__sub__)r.r_)__name__ __module__ __qualname____doc__r]propertyrcrrhr!rjrkrlr3r4rqrrvrzrrrrrrrrrrr"r"r"r#r[|sJ             ,F !  r[c@sXeZdZdZddZeddZeddZdd Zd d Z d d Z ddZ ddZ dS)FiniteFourierSeriesaRepresents 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 cst|}t|}t|}t|trt|dks|\}}ddlm|tfdd|D}|jddddd\}}|dt |d|d d} t d d d d d gd} t dfdd gd} i} i} |D]L}| | t | t | }| | t| t | }|r|| | || tj| || <qj|r|| | || tj| || <qj||7}qjt|| | }t||||S)Nr.rTR10csg|]}|qSr"r")r{r|rr"r#r~sz/FiniteFourierSeries.__new__..F)trig power_base power_explogrrr@cSrFr7rGrIr"r"r#rJrKz-FiniteFourierSeries.__new__..cSs |tjuSr7rLrIr"r"r#rJrMrNrAcrPr7r8rIrQr"r#rJrM)r r/rr)rSrRrrexpandrrrr=r rr getrrrr])r^rUrr9cerexprr!exp_lsrr@rArjrkprYqr")rrr#r]s0  $$  zFiniteFourierSeries.__new__cCsB|jrdnd}|tt|jt|jd7}td|Srf)r!maxsetrjkeysunionrkr)rb_lengthr"r"r#rl#s* zFiniteFourierSeries.intervalcCs |j|jSr7)r4r3rar"r"r#rq)s zFiniteFourierSeries.lengthcCsdt||j}}||jvrtd||f||||}|j|||}|||jd|SNrr r rr(r+rzrrcrr<rbrWrr>rr"r"r#r-  zFiniteFourierSeries.shiftxcCsTt||j}}||jvrtd||f||}|j|}|||jd|Sr)r rr(r+rzrcrr<rr"r"r#r8s    zFiniteFourierSeries.scalecCsdt||j}}||jvrtd||f||||}|j|||}|||jd|Srrrr"r"r#rCrzFiniteFourierSeries.scalexcCsb|dkr|jS|j|tjt|t|j|j|j |tjt |t|j|j}|Sr_) r!rjrrrr rrrrkr )rbr_termr"r"r#rNs &&zFiniteFourierSeries._eval_termcCst|tr|t|j|jdddSt|trD|j|jkr"td|j |j }}|j|j ||}|j |j vr;|St||jddSdS)NrF)finiter)r) r/r[rfourier_seriesrcr<rrhr+rrr()rbrrrrcr"r"r#rVs    zFiniteFourierSeries.__add__N) rrrrr]rrlrqrrrrrr"r"r"r#rs&$     rNTc Cs6t|}t||}|d}||jvr|S|r2t|d|dd}t|||\}}|r2t|||Std}|d|dd}|jr||| } || krft |||\} } t ddt f} t ||| | | fS|| krt j} t ddt f} t|||} t ||| | | fSt |||\} } t|||} t ||| | | fS)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 do not 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)r r6r(rrrZrr is_zerorr$rrr[rrr&) rUrrrr is_finiteres_frcenterneg_fr!rjrkr"r"r#rgs6%      r)NT)%rsympy.core.numbersrrsympy.core.symbolrsympy.core.exprrsympy.core.addrsympy.core.containersrsympy.core.singletonrr r sympy.core.sympifyr (sympy.functions.elementary.trigonometricr r rsympy.series.series_classrsympy.series.sequencesrsympy.sets.setsrsympy.utilities.iterablesrr$r&r6rZr[rrr"r"r"r#s0            8^