chapter 10, Laplace, Hankel and Mellin Transforms

from: Mathematical Techniques for Engineers and Scientists

Author(s): Larry C. Andrews, Ronald L. Phillips

Published: 22 April 2003

Chapter DOI: 10.1117/3.467443.ch10

Chapter Page Count: 50 pages

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Front Matter

1. Ordinary Differential Equations

2. Special Functions

3. Matrix Methods and Linear Vector Spaces

4. Vector Analysis

5. Tensor Analysis

6. Complex Variables

7. Complex Integration, Laurent Series, and Residues

8. Fourier Series, Eigenvalue Problems, and Green's Function

9. Fourier and Related Transforms

10. Laplace, Hankel and Mellin Transforms

11. Calculus of Variations

12. Partial Differential Equations

13. Probability and Random Variables

14. Random Processes

15. Applications

Back Matter

Preview

Chapter Contents

10.1 Introduction
10.2 Laplace Transform
10.3 Initial Value Problems
10.4 Hankel Transform
10.5 Mellin Transform
10.6 Applications Involving the Mellin Transform
10.7 Discrete Fourier Transform
10.8 Z-Transform
10.9 Walsh Transform
Suggested Reading
Exercises

Excerpt

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.

10.1 Introduction

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]). ▶ Integral transform: An integral transform is a relation of the form

such that a given function f(t) is transformed into another function F(s) by means of an integral.

We call the function F(s) the transform of f(t), and K(s,t) is the kernel of the transformation. If the kernel is defined by

where U(t) is the unit step function (Section 2.2), we obtain the Laplace transform (Section 10.2), and if the kernel takes on one of the forms K(s,t) = e^±ist, we obtain the Fourier transform (Chapter 9). Also, if K(s,t) = tJ₀(st)U(t), where J₀(x) is a Bessel function (Section 2.5), we are led to the Hankel transform (Section 10.4), and the Mellin transform (Section 10.5) is associated with the kernel K(s,t) = t^s−1 U(t).

10.2 Laplace Transform

Whereas the Fourier transform grew out of the pioneering work on heat conduction by Joseph Fourier (1768–1830), the Laplace transform can be traced back to the work of Oliver Heaviside (1850–1925) who used operational calculus to solve circuit problems with discontinuous input functions. The Laplace transform is ordinarily defined outright but it can be instructive to develop it directly from the Fourier integral theorem.

DOI: 10.1117/3.467443.ch10

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