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:

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

Plot the sequence:

Find the {3,5} Fourier coefficient:

Find the coefficient of the general term:

Plot the absolute value of coefficients:

Scope  (4)

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

General Fourier coefficient for a Gaussian function:

General Fourier coefficients for Abs:

Fourier coefficient for a basis exponential function:

Options  (2)

Assumptions  (1)

Specify assumptions on a parameter:

FourierParameters  (1)

Use a nondefault setting for FourierParameters:

Properties & Relations  (4)

FourierCoefficient is defined by an integral:

Compute the exponential Fourier series using the individual coefficients:

FourierCoefficient is the same as InverseFourierSequenceTransform:

Fourier coefficients for basis exponentials:

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_2024_fouriercoefficient, author="Wolfram Research", title="{FourierCoefficient}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierCoefficient.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_fouriercoefficient, organization={Wolfram Research}, title={FourierCoefficient}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierCoefficient.html}, note=[Accessed: 22-December-2024 ]}