3.5.11 Integral Transforms and Related Operations

Laplace Transforms

One-dimensional Laplace transforms.

The Laplace transform of a function is given by . The inverse Laplace transform of is given for suitable by .

Laplace transforms have the property that they turn integration and differentiation into essentially algebraic operations. They are therefore commonly used in studying systems governed by differential equations.

Multidimensional Laplace transforms.

Fourier Transforms

One-dimensional Fourier transforms.

In Mathematica the Fourier transform of a function is by default defined to be . The inverse Fourier transform of is similarly defined as .
In different scientific and technical fields different conventions are often used for defining Fourier transforms. The option FourierParameters in Mathematica allows you to choose any of these conventions you want.

Typical settings for FourierParameters with various conventions.

Fourier sine and cosine transforms.

In some applications of Fourier transforms, it is convenient to avoid ever introducing complex exponentials. Fourier sine and cosine transforms correspond to integrating respectively with and instead of , and using limits 0 and rather than and .

Multidimensional Fourier transforms.

Z Transforms

Z transforms.

The Z transform of a function is given by . The inverse Z transform of is given by the contour integral . Z transforms are effectively discrete analogs of Laplace transforms. They are widely used for solving difference equations, especially in digital signal processing and control theory. They can be thought of as producing generating functions, of the kind commonly used in combinatorics and number theory.