# FourierTransform

FourierTransform[expr,t,ω]

gives the symbolic Fourier transform of expr.

FourierTransform[expr,{t1,t2,},{ω1,ω2,}]

gives the multidimensional Fourier transform of expr.

# Details and Options • The Fourier transform of a function is by default defined to be .
• The multidimensional Fourier transform of a function is by default defined to be .
• Other definitions are used in some scientific and technical fields.
• Different choices of definitions can be specified using the option FourierParameters.
• With the setting FourierParameters->{a,b} the Fourier transform computed by FourierTransform is .
• Some common choices for {a,b} are {0,1} (default; modern physics), {1,-1} (pure mathematics; systems engineering), {-1,1} (classical physics), and {0,-2Pi} (signal processing).
• The following options can be given:
•  Assumptions \$Assumptions assumptions to make about parameters FourierParameters {0,1} parameters to define the Fourier transform GenerateConditions False whether to generate answers that involve conditions on parameters
• FourierTransform[expr,t,ω] yields an expression depending on the continuous variable ω that represents the symbolic Fourier transform of expr with respect to the continuous variable t. Fourier[list] takes a finite list of numbers as input, and yields as output a list representing the discrete Fourier transform of the input.
• In TraditionalForm, FourierTransform is output using . »

# Examples

open allclose all

## Scope(6)

Elementary functions:

Special functions:

Piecewise functions and distributions:

Periodic functions:

Multivariate functions:

## Options(3)

### Assumptions(1)

The Fourier transform of BesselJ is a piecewise function:

### FourierParameters(1)

Default modern physics convention:

Convention for pure mathematics, systems engineering:

Convention for classical physics:

Convention for signal processing:

### GenerateConditions(1)

Use to get parameter conditions for when a result is valid:

## Applications(4)

The power spectrum of a damped sinusoid:

The Fourier transform of a radially symmetric function in the plane can be expressed as a Hankel transform. Verify this relation for the function defined by:

Plot the function:

Compute its Fourier transform:

Obtain the same result using HankelTransform:

Plot the Fourier transform:

Generate a gallery of Fourier transforms for a list of radially symmetric functions:

Compute the Hankel transforms for these functions:

Generate the gallery of Fourier transforms as required:

Calculate power spectrum of a stationary OrnsteinUhlenbeckProcess:

## Properties & Relations(4)

Use Asymptotic to compute an asymptotic approximation:

FourierTransform and InverseFourierTransform are mutual inverses:

FourierTransform and FourierCosTransform are equal for even functions:

FourierTransform and FourierSinTransform differ by for odd functions:

## Possible Issues(1)

The result from an inverse Fourier transform may not have the same form as the original:

## Neat Examples(1)

The Fourier transforms of weighted Hermite polynomials have a very simple form:

Introduced in 1999
(4.0)