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FourierCoefficient[expr,t,n]

gives the n^(th) coefficient in the Fourier series expansion of expr.

FourierCoefficient[expr,{t1,t2,…},{n1,n2,…}]

gives a multidimensional Fourier coefficient.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Assumptions  
FourierParameters  
Properties & Relations  
See Also
Related Guides
History
Cite this Page

FourierCoefficient

FourierCoefficient[expr,t,n]

gives the n^(th) coefficient in the Fourier series expansion of expr.

FourierCoefficient[expr,{t1,t2,…},{n1,n2,…}]

gives a multidimensional Fourier coefficient.

Details and Options

  • The ^(th) coefficient in the Fourier series expansion of is by default given by .
  • The -dimensional Fourier coefficient is given by .
  • In the form FourierCoefficient[expr,t,n], n can be symbolic or an integer.
  • The following options can be given:
  • Assumptions $Assumptionsassumptions on parameters
    FourierParameters {1,1}parameters to define Fourier series
    GenerateConditionsFalsewhether to generate results that involve conditions on parameters
  • The function expr is assumed to be periodic in t with period , except when otherwise specified by FourierParameters.
  • Common settings for FourierParameters include:
  • {1,1}f(t) e-i n td tdefault settings
    {1,-2Pi}f(t) ei 2π n td tperiod 1
    {a,b}general setting

Examples

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Basic Examples  (2)

Find the 5^(th) Fourier coefficient:

Wolfram Language code: FourierCoefficient[t ^ 2 + t, t, 5]

Find the coefficient of the general term in a Fourier series:

Wolfram Language code: FourierCoefficient[t ^ 2 + t, t, n]

Plot the sequence:

Wolfram Language code: DiscretePlot[Abs[%], {n, -10, 10}, PlotRange -> All]//Quiet

Find the {3,5} Fourier coefficient:

Wolfram Language code: FourierCoefficient[E ^ (-x - Abs[y]), {x, y}, {3, 5}]

Find the coefficient of the general term:

Wolfram Language code: FourierCoefficient[E ^ (-x - Abs[y]), {x, y}, {m, n}]

Plot the absolute value of coefficients:

Wolfram Language code: ListPointPlot3D[Abs[Table[%, {m, -5, 5}, {n, -5, 5}]], Filling -> Bottom, PlotRange -> All, DataRange -> {{-5, 5}, {-5, 5}}]

Scope  (4)

Find the 3^(rd) Fourier coefficient for an exponential function:

Wolfram Language code: FourierCoefficient[E ^ (-t), t, 3]

General Fourier coefficient for a Gaussian function:

Wolfram Language code: FourierCoefficient[E ^ (-4t ^ 2), t, n]
Wolfram Language code: DiscretePlot[%, {n, -15, 15}, Ticks -> {Automatic, None}]

General Fourier coefficients for Abs:

Wolfram Language code: FourierCoefficient[Abs[t], t, n]

Fourier coefficient for a basis exponential function:

Wolfram Language code: FourierCoefficient[E ^ (3 I t), t, n]
Wolfram Language code: Table[%, {n, -4, 4}]

Options  (2)

Assumptions  (1)

Specify assumptions on a parameter:

Wolfram Language code: FourierCoefficient[E ^ (I a t), t, n, Assumptions -> Element[a, Integers]]

FourierParameters  (1)

Use a nondefault setting for FourierParameters:

Wolfram Language code: Table[FourierCoefficient[t ^ 3, t, n, FourierParameters -> ps], {ps, {{1, 1}, {1, -2π}}}]

Properties & Relations  (4)

FourierCoefficient is defined by an integral:

Wolfram Language code: FourierCoefficient[t ^ 2 + t, t, 3]
Wolfram Language code: 1 / (2π) Integrate[(t ^ 2 + t)E ^ (-3I t), {t, -π, π}]

Compute the exponential Fourier series using the individual coefficients:

Wolfram Language code: FourierSeries[t ^ 2 + t, t, 2]
Wolfram Language code: Sum[FourierCoefficient[(t ^ 2 + t), t, n] E ^ (I n t), {n, -2, 2}]

FourierCoefficient is the same as InverseFourierSequenceTransform:

Wolfram Language code: FourierCoefficient[Abs[t], t, n]
Wolfram Language code: InverseFourierSequenceTransform[Abs[t], t, n]

Fourier coefficients for basis exponentials:

Wolfram Language code: FourierCoefficient[E ^ (-5 I t), t, n]
Wolfram Language code: FourierCoefficient[E ^ (5 I t), t, n]
Wolfram Language code: FourierCoefficient[1, t, n]
Wolfram Research (2008), FourierCoefficient, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierCoefficient.html.

Text

Wolfram Research (2008), FourierCoefficient, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierCoefficient.html.

CMS

Wolfram Language. 2008. "FourierCoefficient." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierCoefficient.html.

APA

Wolfram Language. (2008). FourierCoefficient. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierCoefficient.html

BibTeX

@misc{reference.wolfram_2026_fouriercoefficient, author="Wolfram Research", title="{FourierCoefficient}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierCoefficient.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_fouriercoefficient, organization={Wolfram Research}, title={FourierCoefficient}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierCoefficient.html}, note=[Accessed: 13-June-2026]}