--- title: "FourierCoefficient" language: "en" type: "Symbol" summary: "FourierCoefficient[expr, t, n] gives the n\\[Null]^th coefficient in the Fourier series expansion of expr. FourierCoefficient[expr, {t1, t2, ...}, {n1, n2, ...}] gives a multidimensional Fourier coefficient." keywords: - Fourier series - Fourier expansion - Fourier exponential series - Fourier exponential expansion - orthogonal function series - orthogonal function expansion - Joseph Fourier - boundary value problem - Sturm-Liouville problem canonical_url: "https://reference.wolfram.com/language/ref/FourierCoefficient.html" source: "Wolfram Language Documentation" related_guides: - title: "Fourier Analysis" link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md" - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" related_functions: - title: "FourierSeries" link: "https://reference.wolfram.com/language/ref/FourierSeries.en.md" - title: "FourierSinCoefficient" link: "https://reference.wolfram.com/language/ref/FourierSinCoefficient.en.md" - title: "Fourier" link: "https://reference.wolfram.com/language/ref/Fourier.en.md" - title: "FourierTransform" link: "https://reference.wolfram.com/language/ref/FourierTransform.en.md" - title: "SeriesCoefficient" link: "https://reference.wolfram.com/language/ref/SeriesCoefficient.en.md" - title: "Integrate" link: "https://reference.wolfram.com/language/ref/Integrate.en.md" --- # FourierCoefficient FourierCoefficient[expr, t, n] gives the n$$^{\text{th}}$$ coefficient in the Fourier series expansion of expr. FourierCoefficient[expr, {t1, t2, …}, {n1, n2, …}] gives a multidimensional Fourier coefficient. ## Details and Options * The $n$$$^{\text{th}}$$ coefficient in the Fourier series expansion of $f(t)$ is by default given by $\frac{1}{2\pi }\int _{-\pi }^{\pi }f(t) e^{-i n t}dt$. * The $m$-dimensional Fourier coefficient is given by $\frac{1}{(2\pi )^m}\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }\cdots f\left(t_1,t_2,\ldots \right) e^{-i \left(n_1 t_1+n_2 t_2+\cdots \right)}dt_1dt_2\cdots$. * In the form ``FourierCoefficient[expr, t, n]``, ``n`` can be symbolic or an integer. * The following options can be given: | | | | | ------------------ | ------------- | ----------------------------------------------------------------- | | Assumptions | \$Assumptions | assumptions on parameters | | FourierParameters | {1, 1} | parameters to define Fourier series | | GenerateConditions | False | whether to generate results that involve conditions on parameters | * The function ``expr`` is assumed to be periodic in ``t`` with period $2 \pi$, except when otherwise specified by ``FourierParameters``. * Common settings for ``FourierParameters`` include: | | | | | --- | --- | --- | | {1, 1} | (1/2π)∫-ππf(t) e^-i n td t | default settings | | {1, -2Pi} | ∫-(1/2)(1/2)f(t) e^i 2π n td t | period 1 | | {a, b} | $\left\| \frac{b}{2 \pi }\right\| ^{\frac{a+1}{2}}\int _{-\frac{\pi }{\| b\| }}^{\frac{\pi }{\| b\| }}f(t) e^{-i b n t}dt$ | general setting | --- ## Examples (12) ### Basic Examples (2) Find the 5$$^{\text{th}}$$ Fourier coefficient: ```wl In[1]:= FourierCoefficient[t ^ 2 + t, t, 5] Out[1]= -(2/25) - (I/5) ``` Find the coefficient of the general term in a Fourier series: ```wl In[2]:= FourierCoefficient[t ^ 2 + t, t, n] Out[2]= Piecewise[{{Pi^2/3, n == 0}}, ((-1)^n*(2 + I*n))/n^2] ``` Plot the sequence: ```wl In[3]:= DiscretePlot[Abs[%], {n, -10, 10}, PlotRange -> All]//Quiet Out[3]= [image] ``` --- Find the ``{3, 5}`` Fourier coefficient: ```wl In[1]:= FourierCoefficient[E ^ (-x - Abs[y]), {x, y}, {3, 5}] Out[1]= -(((1/260) - (3 I/260)) E^-π (1 + E^π) Sinh[π]/π^2) ``` Find the coefficient of the general term: ```wl In[2]:= FourierCoefficient[E ^ (-x - Abs[y]), {x, y}, {m, n}] Out[2]= ((-1)^m E^-π ((-1)^1 + n + E^π) Sinh[π]/(1 + I m) (1 + n^2) π^2) ``` Plot the absolute value of coefficients: ```wl In[3]:= ListPointPlot3D[Abs[Table[%, {m, -5, 5}, {n, -5, 5}]], Filling -> Bottom, PlotRange -> All, DataRange -> {{-5, 5}, {-5, 5}}] Out[3]= [image] ``` ### Scope (4) Find the 3$$^{\text{rd}}$$ Fourier coefficient for an exponential function: ```wl In[1]:= FourierCoefficient[E ^ (-t), t, 3] Out[1]= -(((1/10) - (3 I/10)) Sinh[π]/π) ``` --- General Fourier coefficient for a Gaussian function: ```wl In[1]:= FourierCoefficient[E ^ (-4t ^ 2), t, n] Out[1]= (I E^-(n^2/16) (Erfi[(n/4) - 2 I π] - Erfi[(n/4) + 2 I π])/8 Sqrt[π]) In[2]:= DiscretePlot[%, {n, -15, 15}, Ticks -> {Automatic, None}] Out[2]= [image] ``` --- General Fourier coefficients for ``Abs`` : ```wl In[1]:= FourierCoefficient[Abs[t], t, n] Out[1]= Piecewise[{{Pi/2, n == 0}}, -((-1 + (-1)^n)^2/(2*n^2*Pi))] ``` --- Fourier coefficient for a basis exponential function: ```wl In[1]:= FourierCoefficient[E ^ (3 I t), t, n] Out[1]= DiscreteDelta[-3 + n] In[2]:= Table[%, {n, -4, 4}] Out[2]= {0, 0, 0, 0, 0, 0, 0, 1, 0} ``` ### Options (2) #### Assumptions (1) Specify assumptions on a parameter: ```wl In[1]:= FourierCoefficient[E ^ (I a t), t, n, Assumptions -> Element[a, Integers]] Out[1]= DiscreteDelta[-a + n] ``` #### FourierParameters (1) Use a nondefault setting for ``FourierParameters`` : ```wl In[1]:= Table[FourierCoefficient[t ^ 3, t, n, FourierParameters -> ps], {ps, {{1, 1}, {1, -2π}}}] Out[1]= {(I (-1)^n (-6 + n^2 π^2) Sign[n]^4/n^3), -(I (-1)^n (-6 + n^2 π^2) Sign[n]^4/8 n^3 π^3)} ``` ### Properties & Relations (4) ``FourierCoefficient`` is defined by an integral: ```wl In[1]:= FourierCoefficient[t ^ 2 + t, t, 3] Out[1]= -(2/9) - (I/3) In[2]:= 1 / (2π) Integrate[(t ^ 2 + t)E ^ (-3I t), {t, -π, π}] Out[2]= -(2/9) - (I/3) ``` --- Compute the exponential Fourier series using the individual coefficients: ```wl In[1]:= FourierSeries[t ^ 2 + t, t, 2] Out[1]= (-2 + I) E^-I t - (2 + I) E^I t + ((1/2) - (I/2)) E^-2 I t + ((1/2) + (I/2)) E^2 I t + (π^2/3) In[2]:= Sum[FourierCoefficient[(t ^ 2 + t), t, n] E ^ (I n t), {n, -2, 2}] Out[2]= (-2 + I) E^-I t - (2 + I) E^I t + ((1/2) - (I/2)) E^-2 I t + ((1/2) + (I/2)) E^2 I t + (π^2/3) ``` --- ``FourierCoefficient`` is the same as ``InverseFourierSequenceTransform``: ```wl In[1]:= FourierCoefficient[Abs[t], t, n] Out[1]= \[Piecewise]| | | | :------------------------- | :---- | | (π/2) | n == 0 | | -((-1 + (-1)^n)^2/2 n^2 π) | True | In[2]:= InverseFourierSequenceTransform[Abs[t], t, n] Out[2]= \[Piecewise]| | | | :------------------------- | :---- | | (π/2) | n == 0 | | -((-1 + (-1)^n)^2/2 n^2 π) | True | ``` --- Fourier coefficients for basis exponentials: ```wl In[1]:= FourierCoefficient[E ^ (-5 I t), t, n] Out[1]= DiscreteDelta[5 + n] In[2]:= FourierCoefficient[E ^ (5 I t), t, n] Out[2]= DiscreteDelta[-5 + n] In[3]:= FourierCoefficient[1, t, n] Out[3]= DiscreteDelta[n] ``` ## See Also * [`FourierSeries`](https://reference.wolfram.com/language/ref/FourierSeries.en.md) * [`FourierSinCoefficient`](https://reference.wolfram.com/language/ref/FourierSinCoefficient.en.md) * [`Fourier`](https://reference.wolfram.com/language/ref/Fourier.en.md) * [`FourierTransform`](https://reference.wolfram.com/language/ref/FourierTransform.en.md) * [`SeriesCoefficient`](https://reference.wolfram.com/language/ref/SeriesCoefficient.en.md) * [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md) ## Related Guides * [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md) * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) ## History * [Introduced in 2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md)