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$Assumptionsassumptions to make about parameters
    FourierParameters{0,1}parameters to define the Fourier transform
    GenerateConditionsFalsewhether 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

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Basic Examples  (2)

Scope  (6)

Elementary functions:

Special functions:

Piecewise functions and distributions:

Periodic functions:

Multivariate functions:

TraditionalForm formatting:

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 GenerateConditions->True 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)