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# ifourier

Inverse Fourier transform

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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(1/(w^2 + 1), 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.