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fourier

Fourier transform

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

```fourier(f, t, w)
```

Description

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.

Examples

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:

`Pref::fourierParameters(NIL):`

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)`

Parameters

 f Arithmetical expression or unevaluated function call of type fourier. If the first argument is a matrix, the result is returned as a matrix. t Identifier or indexed identifier representing the transformation variable w Arithmetical expression representing the evaluation point

Return Values

Arithmetical expression or matrix of such expressions

f

References

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