Built-in Mathematica Symbol

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.

MORE INFORMATION

The n 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}		default settings
	{1, -2Pi}		period 1
	{a, b}		general setting

EXAMPLES

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Basic Examples (2)

Find the 5

Fourier coefficient:

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

Plot the sequence:

Find the

Fourier coefficient:

Find the coefficient of the general term:

Plot the absolute value of coefficients:

Find the 5

Fourier coefficient:

In[1]:=

Out[1]=

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

Plot the sequence:

Find the

Fourier coefficient:

In[1]:=

Out[1]=

Find the coefficient of the general term:

In[2]:=

Out[2]=

Plot the absolute value of coefficients:

In[3]:=

Out[3]=

Scope (4)

Find the 3

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)

Specify assumptions on a parameter:

Use a non-default 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: