Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 15Integral Transforms

Integral transforms are among the most versatile mathematical tools. Their applications range from solutions of differential and integral equations to evaluation of definite integrals. They can even be used to define fractional derivatives and integrals. In this chapter, after a general introduction, we discuss the two of the most frequently used integral transforms, the Fourier and the Laplace transforms and introduce their properties and techniques. We also discuss discrete Fourier transforms and the fast Fourier transform in detail. We finally introduce the Mellin and the Radon transforms, where the latter has some interesting applications to medical technology and CAT scanning.

15.1 Some Commonly Encountered Integral Transforms

Commonly encountered integral transforms allow us to relate two functions through the integral

15.1

where is called the integral transform of with respect to the kernel . These transformations are linear, that is, if the transforms and :

15.2

exist, then one can write

15.3

and

15.4

where is a constant. Integral transforms ...