# 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 t default settings {1,-2Pi} f(t) ei 2π n td t 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 {3,5} Fourier coefficient:

Find the coefficient of the general term:

Plot the absolute value of coefficients:

## 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)

### 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:

Introduced in 2008
(7.0)