Delta Functions

Introduction to Generalised Functions

R F Hoskins

• Publisher: Woodhead Publishing
• Year: 2009
• ISBN:
  9780857099358
• Edition: 2
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Delta Functions has now been updated, restructured and modernised into a second edition, to answer specific difficulties typically found by students encountering delta functions for the first time. In particular, the treatment of the Laplace transform has been revised with this in mind. The chapter on Schwartz distributions has been considerably extended and the book is supplemented by a fuller review of Nonstandard Analysis and a survey of alternative infinitesimal treatments of generalised functions.

Dealing with a difficult subject in a simple and straightforward way, the text is readily accessible to a broad audience of scientists, mathematicians and engineers. It can be used as a working manual in its own right, and serves as a preparation for the study of more advanced treatises. Little more than a standard background in calculus is assumed, and attention is focused on techniques, with a liberal selection of worked examples and exercises.

• Second edition has been updated, restructured and modernised to answer specific difficulties typically found by students encountering delta functions for the first time
• Attention is focused on techniques, with a liberal selection of worked examples and exercises
• Readily accessible to a broad audience of scientists, mathematicians and engineers and can be used as a working manual in its own right

Chapter 1

Results from Elementary Analysis

1.1 THE REAL NUMBER SYSTEM

1.1.1 Introduction.

The term ‘analysis’ which appears in the heading of this chapter deserves some preliminary comment. It may loosely be described as that branch of mathematics which is concerned with the concepts and processes which underlie the calculus, more especially with the nature of number, function and limit. A clear understanding of such concepts is necessary for the study of advanced calculus and the theory of functions of a real variable. Inevitably this involves re-examining many of the ideas and arguments which are usually taken as intuitively obvious in a first course in calculus, and many students (particularly those whose primary interest is in mathematics as a tool) find this exacting and difficult to justify. One such student was once heard to give a disparaging definition of analysis as ‘a sort of revision course in calculus, only done much more slowly’.

It is not our purpose to offer such a course here. Indeed, little more than a standard background in the techniques of elementary calculus is required for the greater part of this book, and most of the material in this chapter should already be familia...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright page
• About the Author
• Preface
• Chapter 1: Results from Elementary Analysis
• Chapter 2: The Dirac Delta Function
• Chapter 3: Properties of the Delta Function
• Chapter 4: Time-invariant Linear Systems
• Chapter 5: The Laplace Transform
• Chapter 6: Fourier Series and Transforms
• Chapter 7: Other Generalised Functions
• Chapter 8: Introduction to distributions
• Chapter 9: Integration Theory
• Chapter 10: Introduction to N.S.A
• Chapter 11: Nonstandard Generalised Functions
• Solutions to Exercises
• Index