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FourierSinCoefficient[expr,t,n]

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

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

gives a multidimensional Fourier sine coefficient.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
See Also
Related Guides
History
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FourierSinCoefficient

FourierSinCoefficient[expr,t,n]

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

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

gives a multidimensional Fourier sine coefficient.

Details and Options

  • The ^(th) coefficient in the Fourier sine series expansion of is by default given by .
  • The -dimensional Fourier sine coefficient is given by .
  • In the form FourierSinCoefficient[expr,t,n], n can be symbolic or a positive 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 3^(rd) Fourier sine series coefficient:

Find the general term coefficient:

Plot the coefficient sequence:

Find the {3,5} Fourier sine coefficient:

The general term:

Plot the coefficient sequence:

Scope  (4)

Find the 5^(th) Fourier sine coefficient for a quadratic polynomial:

Find the general coefficient for a piecewise function:

General Fourier sine coefficient for a Gaussian function:

Fourier sine coefficient for a basis function:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_fouriersincoefficient, author="Wolfram Research", title="{FourierSinCoefficient}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierSinCoefficient.html}", note=[Accessed: 08-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_fouriersincoefficient, organization={Wolfram Research}, title={FourierSinCoefficient}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierSinCoefficient.html}, note=[Accessed: 08-August-2025]}