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# `ifourier`

Inverse Fourier transform

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

## Syntax

```ifourier(`F`, `w`, `t`)
```

## Description

`ifourier(F, w, t)` computes the inverse Fourier transform of the expression `F = F(w)` with respect to the variable `w` at the point `t`.

The inverse Fourier transform of the expression ```F = F(w)``` with respect to the variable `w` at the point `t` 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 `F` is a matrix, `ifourier` applies the inverse Fourier transform to all components of the matrix.

MuPAD® computes `ifourier(F, w, t)` as

.

If `ifourier` cannot find an explicit representation of the inverse Fourier transform, it returns results in terms of the direct Fourier transform. See Example 4.

To compute the direct Fourier transform, use `fourier`.

To compute the inverse discrete Fourier transform, use `numeric::invfft`.

## Environment Interactions

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

## Examples

### Example 1

Compute the inverse Fourier transform of this expression with respect to the variable `w`:

`ifourier(sqrt(PI)*exp(-w^2/4), w, t)`

### Example 2

Compute the inverse Fourier transform of this expression with respect to the variable `w` for positive values of the parameter t0:

```assume(t_0 > 0): f := ifourier(-(PI^(1/2)*w*exp(-w^2*t_0^2/4)*I)*t_0^3/2, w, t)```

Evaluate the inverse Fourier transform of the expression at the points t = - 2 t0 and t = 1. You can evaluate the resulting expression `f` using `|` (or its functional form `evalAt`):

`f | t = -2*t_0`

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

`ifourier(-(PI^(1/2)*w*exp(-w^2*t_0^2/4)*I)*t_0^3/2, w, 1)`

### Example 3

The default parameters of the Fourier and inverse Fourier transforms are `c = 1` and `s = -1`:

`ifourier(-(sqrt(PI)*w*exp(-w^2/4)*I)/2, w, t)`

To change these parameters, use `Pref::fourierParameters` before calling `ifourier`:

`Pref::fourierParameters(1, 1):`

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

`ifourier(-(sqrt(PI)*w*exp(-w^2/4)*I)/2, w, t)`

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

`Pref::fourierParameters(NIL):`

### Example 4

If `ifourier` cannot find an explicit representation of the transform, it returns results in terms of the direct Fourier transform:

`ifourier(exp(-w^4), w, t)`

### Example 5

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

`ifourier(dirac(w), w, t)`

`ifourier(heaviside(w + 5), w, t)`

## Parameters

 `F` Arithmetical expression or matrix of such expressions `w` Identifier or indexed identifier representing the transformation variable `t` Arithmetical expression representing the evaluation point

## Return Values

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

`F`

## References

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

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