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**MuPAD® notebooks will be removed in a future release. 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.**

There are several commonly used conventions for defining Fourier
transforms. MuPAD^{®} defines the Fourier transform (FT) as:

Here `c`

and `s`

are the parameters
of the Fourier transform. By default, `c = 1`

and `s = -1`

. `Pref::fourierParameters`

lets
you specify other values for these parameters. For the inverse Fourier
transform (IFT), MuPAD uses the following definition:

To compute the Fourier transform of an arithmetical expression,
use the `fourier`

function.
For example, compute the Fourier transforms of the following exponential
expression and the Dirac delta distribution:

fourier(exp(-t^2), t, w), fourier(dirac(t), t, w)

If you know the Fourier transform of an expression, you can
find the original expression or its mathematically equivalent form
by computing the inverse Fourier transform. To compute the inverse
Fourier transform, use the `ifourier`

function.
For example, find the original exponential expression and the Dirac
delta distribution:

ifourier(PI^(1/2)*exp(-w^2/4), w, t), ifourier(1, w, t)

Suppose, you compute the Fourier transform of an expression, and then compute the inverse Fourier transform of the result. In this case, MuPAD can return an expression that is mathematically equivalent to the original one, but presented in a different form. For example, compute the Fourier transforms of the following trigonometric expressions:

Cosine := fourier(cos(t), t, w); Sine := fourier(sin(t^2), t, w)

Now, compute the inverse Fourier transforms of the resulting
expressions `Cosine`

and `Sine`

.
The results differ from the original expressions:

invCosine := ifourier(Cosine, w, t); invSine := ifourier(Sine, w, t)

Simplifying the resulting expressions `invCosine`

and `invSine`

gives
the original expressions:

simplify(invCosine), simplify(invSine)

Besides arithmetical expressions, the `fourier`

and `ifourier`

functions also accept matrices
of arithmetical expressions. For example, compute the Fourier transform
of the following matrix:

A := matrix(2, 2, [exp(-t^2), t*exp(-t^2), t^2*exp(-t^2), t^3*exp(-t^2)]): fourier(A, t, w)

The `fourier`

and `ifourier`

functions let
you evaluate the transforms of an expression or a matrix at a particular
point. For example, evaluate the Fourier transform of the matrix `A`

for
the values `w = 0`

and `w = 2*x`

:

fourier(A, t, 0); fourier(A, t, 2*x)

If MuPAD cannot compute the Fourier transform of an expression, it returns an unresolved transform:

fourier(f(t), t, w)

If MuPAD cannot compute the inverse Fourier transform of an expression, it returns the result in terms of an unresolved direct Fourier transform:

ifourier(F(w), w, t)

The Laplace transform is defined as follows:

.

The inverse Laplace transform is defined by a contour integral in the complex plane:

,

where *c* is
a real value. To compute the Laplace transform of an arithmetical
expression, use the `laplace`

function.
For example, compute the Laplace transform of the following expression:

tsine := laplace(t*sin(a*t), t, s)

To compute the original expression from its Laplace transform,
perform the inverse Laplace transform. To compute the inverse Laplace
transform, use the `ilaplace`

function.
For example, compute the inverse Laplace transform of the resulting
expression `tsine`

:

ilaplace(tsine, s, t)

Suppose, you compute the Laplace transform of an expression, and then compute the inverse Laplace transform of the result. In this case, MuPAD can return an expression that is mathematically equivalent to the original one, but presented in a different form. For example, compute the Laplace transforms of the following expression:

L := laplace(t*ln(t), t, s)

Now, compute the inverse Laplace transform of the resulting
expression `L`

. The result differs from the original
expression:

invL := ilaplace(L, s, t)

Simplifying the expression `invL`

gives the
original expression:

simplify(invL)

Besides arithmetical expressions, the `laplace`

and `ilaplace`

functions also accept matrices
of arithmetical expressions. For example, compute the Laplace transform
of the following matrix:

A := matrix(2, 2, [1, t, t^2, t^3]): laplace(A, t, s)

When computing a transform of an expression, you can use assumptions on mathematical properties of the arguments. For example, compute the Laplace transform of the Dirac delta distribution:

d := laplace(dirac(t - t_0), t, s) assuming t_0 >=0

Restore the Dirac delta distribution from the resulting expression `d`

:

ilaplace(d, s, t) assuming t_0 >=0

The `laplace`

function
provides the transforms for some special functions. For example, compute
the Laplace transforms of the following Bessel functions:

laplace(besselJ(0, t), t, s); laplace(besselJ(1, t), t, s); laplace(besselJ(1/2, t), t, s)

The `laplace`

and `ilaplace`

functions let
you evaluate the transforms of an expression or a matrix at a particular
point. For example, evaluate the Laplace transform of the following
expression for the value `s = 10`

:

laplace(t*exp(-t), t, 10)

Now, evaluate the inverse Laplace transform of the following
expression for the value `t = x + y`

:

ilaplace(1/(1 + s)^2, s, x + y)

If MuPAD cannot compute the Laplace transform or the inverse Laplace transform of an expression, it returns an unresolved transform:

laplace(f(t), t, s)

ilaplace(F(s), s, t)