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FourierTrigSeries[expr,t,n]

gives the n^(th)-order Fourier trigonometric series expansion of expr in t.

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

gives the multidimensional Fourier trigonometric series of expr.

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

FourierTrigSeries

FourierTrigSeries[expr,t,n]

gives the n^(th)-order Fourier trigonometric series expansion of expr in t.

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

gives the multidimensional Fourier trigonometric series of expr.

Details and Options

  • The n^(th)-order Fourier trigonometric series of is by default defined to be with and .
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    FourierParameters {1,1}parameters to define Fourier trig series
    GenerateConditionsFalsewhether to generate results that involve conditions on parameters
  • With the setting FourierParameters->{a,b} the following series is returned: with and .

Examples

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Basic Examples  (2)

Find the 5^(th)-order Fourier trigonometric series of t:

Wolfram Language code: FourierTrigSeries[t, t, 5]
Wolfram Language code: Plot[%, {t, -3π, 3π}]

Find the 3^(rd)-order bivariate Fourier trigonometric series approximation to :

Wolfram Language code: FourierTrigSeries[x ^ 2 y, {x, y}, {3, 3}]
Wolfram Language code: Plot3D[%, {x, -Pi, Pi}, {y, -Pi, Pi}]

Scope  (4)

Find the Fourier trigonometric series of an exponential function:

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

Fourier trigonometric series for a Gaussian function:

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

Fourier trigonometric series for Abs:

Wolfram Language code: FourierTrigSeries[Abs[t], t, 2]
Wolfram Language code: Plot[{%, Abs[t]}, {t, -Pi, Pi}]

The Fourier trigonometric series for a basis function has only one term:

Wolfram Language code: FourierTrigSeries[Cos[3 t], t, 5]
Wolfram Language code: FourierTrigSeries[Sin[3 t], t, 5]

Options  (1)

FourierParameters  (1)

Use a nondefault setting for FourierParameters:

Wolfram Language code: FourierTrigSeries[E ^ (-Abs[t]), t, 2, FourierParameters -> {1, -2π}]
Wolfram Research (2008), FourierTrigSeries, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierTrigSeries.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_fouriertrigseries, author="Wolfram Research", title="{FourierTrigSeries}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierTrigSeries.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_fouriertrigseries, organization={Wolfram Research}, title={FourierTrigSeries}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierTrigSeries.html}, note=[Accessed: 13-June-2026]}