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Integral Transforms

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Fourier and Inverse Fourier Transforms

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) Laplace and Inverse Laplace Transforms

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) Mathematical Modeling with Symbolic Math Toolbox

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