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2001 | Buch

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An Introduction to Laplace Transforms and Fourier Series

verfasst von: Philip P. G. Dyke, BSc, PhD

Verlag: Springer London

Buchreihe : Springer Undergraduate Mathematics Series

Enthalten in: Professional Book Archive

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Über dieses Buch

This book has been primarily written for the student of mathematics who is in the second year or the early part of the third year of an undergraduate course. It will also be very useful for students of engineering and the physical sciences for whom Laplace Transforms continue to be an extremely useful tool. The book demands no more than an elementary knowledge of calculus and linear algebra of the type found in many first year mathematics modules for applied subjects. For mathematics majors and specialists, it is not the mathematics that will be challenging but the applications to the real world. The author is in the privileged position of having spent ten or so years outside mathematics in an engineering environment where the Laplace Transform is used in anger to solve real problems, as well as spending rather more years within mathematics where accuracy and logic are of primary importance. This book is written unashamedly from the point of view of the applied mathematician. The Laplace Transform has a rather strange place in mathematics. There is no doubt that it is a topic worthy of study by applied mathematicians who have one eye on the wealth of applications; indeed it is often called Operational Calculus.

Inhaltsverzeichnis

Frontmatter

1. The Laplace Transform

Abstract

As a discipline, mathematics encompasses a vast range of subjects. In pure mathematics an important concept is the idea of an axiomatic system whereby axioms are proposed and theorems are proved by invoking these axioms logically. These activities are often of little interest to the applied mathematician to whom the pure mathematics of algebraic structures will seem like tinkering with axioms for hours in order to prove the obvious. To the engineer, this kind of pure mathematics is even more of an anathema. The value of knowing about such structures lies in the ability to generalise the “obvious” to other areas. These generalisations are notoriously unpredictable and are often very surprising. Indeed, many say that there is no such thing as non-applicable mathematics, just mathematics whose application has yet to be found.

Philip P. G. Dyke

2. Further Properties of the Laplace Transform

Abstract

Sometimes, a function F(t) represents a natural or engineering process that has no obvious starting value. Statisticians call this a time series. Although we shall not be considering F(t) as stochastic, it is nevertheless worth introducing a way of “switching on” a function. Let us start by finding the Laplace Transform of a step function the name of which pays homage to the pioneering electrical engineer Oliver Heaviside (1850–1925). The formal definition runs as follows.

Philip P. G. Dyke

3. Convolution and the Solution of Ordinary Differential Equations

Abstract

It is assumed from the outset that students will have some familiarity with ordinary differential equations (ODE), but there is a brief résumé given in Section 3.3. The other central and probably new idea is that of the convolution integral and this is introduced fully in Section 3.2. Of course it is possible to solve some kinds of differential equation without using convolution as is obvious from the last chapter, but mastery of the convolution theorem greatly extends the power of Laplace Transforms to solve ODEs. In fact, familiarity with the convolution operation is necessary for the understanding of many other topics that feature in this text such as the solution of partial differential equations (PDEs) and other topics that are outside it such as the use of Green’s functions for forming the general solution of various types of boundary value problem (BVP).

Philip P. G. Dyke

4. Fourier Series

Abstract

Before getting to Fourier series proper, we need to discuss the context. To understand why Fourier series are so useful, one would need to define an inner product space and show that trigonometric functions are an example of one. It is the properties of the inner product space, coupled with the analytically familiar properties of the sine and cosine functions that give Fourier series their usefulness and power. Some familiarity with set theory, vector and linear spaces would be useful. These are topics in the first stages of most mathematical degrees, but if they are new, the text by Whitelaw (1983) will prove useful

Philip P. G. Dyke

5. Partial Differential Equations

Abstract

In previous chapters, we have explained how ordinary differential equations can be solved using Laplace Transforms. In Chapter 4, Fourier series were introduced, and the important property that any reasonable function can be expressed as a Fourier series derived. In this chapter, these ideas are brought together, and the solution of certain types of partial differential equation using both Laplace Transforms and Fourier Series are explored. The study of the solution of partial differential equations (abbreviated PDEs) is a vast topic that it is neither possible nor appropriate to cover in a single chapter. There are many excellent texts (Sneddon (1957) and Williams (1980) to name but two) that have become standard. Here we shall only be interested in certain types of PDE that are amenable to solution by Laplace Transform.

Philip P. G. Dyke

6. Fourier Transforms

Abstract

Later in this chapter we define the Fourier Transform. There are two ways of approaching the subject of Fourier Transforms, both ways are open to us! One way is to carry on directly from Chapter 4 and define Fourier Transforms in terms of the mathematics of linear spaces by carefully increasing the period of the function f(x). This would lead to the Fourier series we defined in Chapter 4 becoming, in the limit of infinite period, an integral. This integral leads directly to the Fourier Transform. On the other hand, the Fourier Transform can be straightforwardly defined as an example of an integral transform and its properties compared and in many cases contrasted with those of the Laplace Transform. It is this second approach that is favoured here, with the first more pure mathematical approach outlined towards the end of Section 6.2. This choice is arbitrary, but it is felt that the more “hands on” approach should dominate here. Having said this, texts that concentrate on computational aspects such as the FFT (Fast Fourier Transform), on time series analysis and on other branches of applied statistics sometimes do prefer the more pure approach in order to emphasise precision.

Philip P. G. Dyke

7. Complex Variables and Laplace Transforms

Abstract

The material in this chapter is written on the assumption that you have some familiarity with complex variable theory (or complex analysis). That is we assume that defining f(z) where \(z = x + iy,i = \sqrt { - 1} \) and where x and y are independent variables is not totally mysterious. In Laplace Transforms, s can fruitfully be thought of as a complex variable. Indeed parts of this book (Section 6.2 for example) have already strayed into this territory.

Philip P. G. Dyke

Backmatter

Titel: An Introduction to Laplace Transforms and Fourier Series
verfasst von: Philip P. G. Dyke, BSc, PhD
Verlag: Springer London
Electronic ISBN: 978-1-4471-0505-3
Print ISBN: 978-1-85233-015-6
DOI: https://doi.org/10.1007/978-1-4471-0505-3

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. The Laplace Transform

2. Further Properties of the Laplace Transform

3. Convolution and the Solution of Ordinary Differential Equations

4. Fourier Series

5. Partial Differential Equations

6. Fourier Transforms

7. Complex Variables and Laplace Transforms

Backmatter