coefficient in the Fourier series expansion of 
FourierCoefficient
FourierCoefficient[expr,t,n]
gives the n
coefficient in the Fourier series expansion of expr.
FourierCoefficient[expr,{t1,t2,…},{n1,n2,…}]
gives a multidimensional Fourier coefficient.
Details and Options
- The
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 $Assumptions assumptions on parameters FourierParameters {1,1} parameters to define Fourier series GenerateConditions False whether 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
open all close allBasic Examples (2)
Fourier coefficient: Scope (4)
Find the 3
Fourier coefficient for an exponential function:
General Fourier coefficient for a Gaussian function:
General Fourier coefficients for Abs:
Options (2)
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:
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