In mathematics, a transform is often some expression or process that is used to convert one type of problem into another that is in some sense easier to solve. It must then be possible to reverse the process to obtain the solution of the original problem from the solution of the transformed problem.

In this chapter we define four transforms (Fourier, Fourier sine, Fourier cosine, and Laplace), which are defined in terms of integrals and are well suited to the solution of some types of initial-boundary value problems.

The *Fourier transform *[*f*] of a function *f* is defined to be the function of ω defined by

(7.1)

Because it is awkward to write [*f*](ω) in performing calculations, we often denote the Fourier transform of *f* as . In this notation,

We rarely compute a Fourier transform by integration, but instead refer to a set of tables or a software package.

**Example 7.1** Let *a* be a positive constant and let

When working ...

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