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FourierCosCoefficient[expr,t,n]

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

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

gives a multidimensional Fourier cosine coefficient.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
FourierParameters  
See Also
Related Guides
History
Cite this Page

FourierCosCoefficient

FourierCosCoefficient[expr,t,n]

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

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

gives a multidimensional Fourier cosine coefficient.

Details and Options

  • The ^(th) coefficient in the Fourier cosine series expansion of is by default given by .
  • The –dimensional Fourier cosine coefficient is given by .
  • In the form FourierCosCoefficient[expr,t,n], n can be symbolic or a non–negative 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}default settings
    {1,2Pi}period 1
    {a,b}general setting

Examples

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Basic Examples  (2)

Find the 5^(th) Fourier cosine coefficient:

Wolfram Language code: FourierCosCoefficient[t ^ 2, t, 5]

Find the general term coefficient:

Wolfram Language code: FourierCosCoefficient[t ^ 2, t, n]

Plot the sequence:

Wolfram Language code: DiscretePlot[%, {n, -7, 7}, PlotRange -> All]//Quiet

Find the {3,5} Fourier cosine coefficient:

Wolfram Language code: FourierCosCoefficient[x ^ 2 y, {x, y}, {3, 5}]

Find the coefficient of the general term:

Wolfram Language code: FourierCosCoefficient[x ^ 2 y, {x, y}, {m, n}]

Plot the multivariate sequence:

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

Scope  (4)

Find the 6^(th) Fourier cosine coefficient for a quadratic polynomial:

Wolfram Language code: FourierCosCoefficient[t ^ 2 + 7t + 2, t, 6]

General Fourier cosine coefficient for a piecewise function:

Wolfram Language code: FourierCosCoefficient[UnitStep[t(Pi / 2 - t)], t, n]

Fourier cosine coefficient for a Gaussian function:

Wolfram Language code: FourierCosCoefficient[E ^ (-t ^ 2), t, n]

Fourier cosine coefficient for a basis function:

Wolfram Language code: FourierCosCoefficient[Cos[3t], t, n]
Wolfram Language code: Table[%, {n, 0, 10}]

Options  (1)

FourierParameters  (1)

Use a nondefault setting for FourierParameters:

Wolfram Language code: Table[FourierCosCoefficient[t, t, n, FourierParameters -> fp], {fp, {{1, 1}, {1, 2π}}}]
Wolfram Research (2008), FourierCosCoefficient, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierCosCoefficient.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_fouriercoscoefficient, author="Wolfram Research", title="{FourierCosCoefficient}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierCosCoefficient.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_fouriercoscoefficient, organization={Wolfram Research}, title={FourierCosCoefficient}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierCosCoefficient.html}, note=[Accessed: 13-June-2026]}