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Chapter 10:
Laplace, Hankel and Mellin Transforms
In this chapter we introduce several transforms that are commonly used in engineering applications. Except for the Laplace transform, which can be used in a variety of applications, the other integral transforms are considered more specialized. We also briefly discuss the notion of a discrete Fourier transform, a discrete Laplace transform (called the Z-transform), and a discrete Walsh transform. Integral transforms are common working tools of every engineer and scientist. The Fourier transform studied in Chapter 9 is basic to frequency spectrum analysis of time-varying waveforms. Here, we study the Laplace transform used in control theory and in the analysis of initial-value problems like those associated with electric circuits. In addition, we introduce the Hankel transform (directly related to a two-dimensional Fourier transform) and the Mellin transform. The Hankel transform is essential to the analysis of diffraction theory and image formation in optics (see [15]), and the Mellin transform is useful in probability theory and in optical wave propagation (see [28]).
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