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Fourier transform

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fourier(f, t, w)


fourier(f, t, w) computes the Fourier transform of the expression f = f(t) with respect to the variable t at the point w and is defined as follows:


c and s are parameters of the Fourier transform. By default, c = 1 and s = -1.

To change the parameters c and s of the Fourier transform, use Pref::fourierParameters. See Example 3. Common choices for the parameter c are 1, , or . Common choices for the parameter s are -1, 1, - 2 π, or 2 π.

If fourier cannot find an explicit representation of the transform, it returns an unevaluated function call. See Example 4.

If f is a matrix, fourier applies the Fourier transform to all components of the matrix.

To compute the inverse Fourier transform, use ifourier.

To compute the discrete Fourier transform, use numeric::fft.

Environment Interactions

Results returned by fourier depend on the current Pref::fourierParameters settings.


Example 1

Compute the Fourier transform of this expression with respect to the variable t:

fourier(exp(-t^2), t, w)

Example 2

Compute the Fourier transform of this expression with respect to the variable t for positive values of the parameter w0:

assume(w_0 > 0):
F := fourier(t*exp(-w_0^2*t^2), t, w)

Evaluate the Fourier transform of the expression at the points w = 2 w0 and w = 5. You can evaluate the resulting expression F using | (or its functional form evalAt):

F | w = 2*w_0

Also, you can evaluate the Fourier transform at a particular point directly:

fourier(t*exp(-w_0^2*t^2), t, 5)

Example 3

The default parameters of the Fourier transform are c = 1 and s = -1.

fourier(t*exp(-t^2), t, w)

To change these parameters, use Pref::fourierParameters before calling fourier:

Pref::fourierParameters(1, 1):

Evaluate the transform of the same expression with the new parameters:

fourier(t*exp(-t^2), t, w)

For further computations, restore the default values of the Fourier transform parameters:


Example 4

If fourier cannot find an explicit representation of the transform, it returns an unevaluated call:

fourier(besselJ(1, 1/(1 + t^2)), t, w)

ifourier returns the original expression:

ifourier(%, w, t)

Example 5

Compute the following Fourier transforms that involve the Dirac and the Heaviside functions:

fourier(t^3, t, w)

fourier(heaviside(t - t_0), t, w)

Example 6

The Fourier transform of a function is related to the Fourier transform of its derivative:

fourier(diff(f(t), t), t, w)



Arithmetical expression or unevaluated function call of type fourier. If the first argument is a matrix, the result is returned as a matrix.


Identifier or indexed identifier representing the transformation variable


Arithmetical expression representing the evaluation point

Return Values

Arithmetical expression or matrix of such expressions

Overloaded By



F. Oberhettinger, “Tables of Fourier Transforms and Fourier Transforms of Distributions”, Springer, 1990.

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