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# fourier

Fourier transform

## Syntax

fourier(f,trans_var,eval_point)

## Description

fourier(f,trans_var,eval_point) computes the Fourier transform of f with respect to the transformation variable trans_var at the point eval_point.

## Input Arguments

 f Symbolic expression, symbolic function, or vector or matrix of symbolic expressions or functions. trans_var Symbolic variable representing the transformation variable. This variable is often called the "time variable" or the "space variable". Default: The variable determined by symvar. eval_point Symbolic variable or expression representing the evaluation point. This variable is often called the "frequency variable". Default: The variable w. If w is the transformation variable of f, then the default evaluation point is the variable v.

## Examples

Compute the Fourier transform of this expression with respect to the variable x at the evaluation point y:

```syms x y
f = exp(-x^2);
fourier(f, x, y)```
```ans =
pi^(1/2)*exp(-y^2/4)```

Compute the Fourier transform of this expression calling the fourier function with one argument. If you do not specify the transformation variable, it is determined by symvar. For this expression, symvar chooses x as the transformation variable.

```syms x t y
f = exp(-x^2)*exp(-t^2);
fourier(f, y)```
```ans =
pi^(1/2)*exp(-t^2)*exp(-y^2/4)```

If you also do not specify the evaluation point, fourier uses the variable w:

`fourier(f)`
```ans =
pi^(1/2)*exp(-t^2)*exp(-w^2/4)```

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

```syms t w
fourier(t^3, t, w)```
```ans =
-pi*dirac(w, 3)*2*i```
```syms t0
fourier(heaviside(t - t0), t, w)```
```ans =
exp(-t0*w*i)*(pi*dirac(w) - i/w)```
```assume(x,'real')
f = exp(-x^2*abs(t))*sin(t)/t;
fourier(f, t, w)```
```ans =
piecewise([x ~= 0, atan((w + 1)/x^2) - atan((w - 1)/x^2)])```

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

```syms f(t) w
F = fourier(f, t, w)```
```F(w) =
fourier(f(t), t, w)```

ifourier returns the original expression:

`ifourier(F, w, t)`
```ans(t) =
f(t)```

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

```syms f(t) w
fourier(diff(f(t), t), t, w)```
```ans =
w*fourier(f(t), t, w)*i```

## More About

expand all

### Fourier Transform

The Fourier transform of the expression f = f(x) with respect to the variable x at the point w is defined as follows:

Here c and s are parameters of the Fourier transform. The fourier function uses c = 1, s = –1.

### Tips

• If you call fourier with two arguments, it assumes that the second argument is the evaluation point eval_point.

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

• To compute the inverse Fourier transform, use ifourier.

## References

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

## See Also

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