# Documentation

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# `fourier`

Fourier transform

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## 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`

## Algorithms

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