Delta Functions

Delta Functions

Introduction to Generalised Functions

2nd Edition - March 31, 2009

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  • Author: R F Hoskins
  • eBook ISBN: 9780857099358

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Description

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.

Key Features

  • 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

Readership

Scientists, mathematicians, and engineers

Table of Contents

  • Chapter 1: Results from Elementary Analysis

    • 1.1 THE REAL NUMBER SYSTEM
    • 1.2 FUNCTIONS
    • 1.3 CONTINUITY
    • 1.4 DIFFERENTIABILITY
    • 1.5 TAYLOR’S THEOREM
    • 1.6 INTEGRATION
    • 1.7 IMPROPER INTEGRALS
    • 1.8 UNIFORM CONVERGENCE
    • 1.9 DIFFERENTIATING INTEGRALS

    Chapter 2: The Dirac Delta Function

    • 2.1 THE UNIT STEP FUNCTION
    • 2.2 DERIVATIVE OF THE UNIT STEP FUNCTION
    • 2.3 THE DELTA FUNCTION AS A LIMIT
    • 2.4 STIELTJES INTEGRALS
    • 2.5 DEVELOPMENTS OF DELTA FUNCTION THEORY
    • 2.6 HISTORICAL NOTE

    Chapter 3: Properties of the Delta Function

    • 3.1 THE DELTA FUNCTION AS A FUNCTIONAL
    • 3.2 SUMS AND PRODUCTS
    • Exercises I
    • 3.3 DIFFERENTIATION
    • Exercises II
    • 3.4 DERIVATIVES OF THE DELTA FUNCTION
    • 3.5 POINTWISE DESCRIPTION OF δ′(t)
    • 3.6 INTEGRATION OF THE DELTA FUNCTION
    • 3.7 CHANGE OF VARIABLE

    Chapter 4: Time-invariant Linear Systems

    • 4.1 SYSTEMS AND OPERATORS
    • 4.2 STEP RESPONSE AND IMPULSE RESPONSE
    • 4.3 CONVOLUTION
    • 4.4 IMPULSE RESPONSE FUNCTIONS
    • 4.5 TRANSFER FUNCTION

    Chapter 5: The Laplace Transform

    • 5.1 THE CLASSICAL LAPLACE TRANSFORM
    • 5.2 LAPLACE TRANSFORMS OF DELTA FUNCTIONS
    • 5.3 COMPUTATION OF LAPLACE TRANSFORMS
    • 5.4 NOTE ON INVERSION

    Chapter 6: Fourier Series and Transforms

    • 6.1 FOURIER SERIES
    • 6.2 GENERALISED FOURIER SERIES
    • 6.3 FOURIER TRANSFORMS
    • 6.4 GENERALISED FOURIER TRANSFORMS

    Chapter 7: Other Generalised Functions

    • 7.1 FRACTIONAL CALCULUS
    • 7.2 HADAMARD FINITE PART
    • 7.3 PSEUDO-FUNCTIONS

    Chapter 8: Introduction to distributions

    • 8.1 TEST FUNCTIONS
    • 8.2 FUNCTIONALS AND DISTRIBUTIONS
    • 8.3 CALCULUS OF DISTRIBUTIONS
    • 8.4 GENERAL SCHWARTZ DISTRIBUTIONS

    Chapter 9: Integration Theory

    • 9.1 RIEMANN-STIELTJES INTEGRALS
    • 9.2 EXTENSION OF THE ELEMENTARY INTEGRAL
    • 9.3 THE LEBESGUE AND RIEMANN INTEGRALS

    Chapter 10: Introduction to N.S.A

    • 10.1 A NONSTANDARD NUMBER SYSTEM
    • 10.2 NONSTANDARD EXTENSIONS
    • 10.3 ELEMENTARY ANALYSIS
    • 10.4 INTERNAL OBJECTS

    Chapter 11: Nonstandard Generalised Functions

    • 11.1 NONSTANDARD δ-FUNCTIONS
    • 11.2 PRE-DELTA FUNCTIONS
    • 11.3 PERIODIC DELTA FUNCTIONS
    • 11.4 N.S.A. AND DISTRIBUTIONS

    Solutions to Exercises

    • CHAPTER 2
    • CHAPTER 3
    • CHAPTER 4
    • CHAPTER 5
    • CHAPTER 6

    Index

Product details

  • No. of pages: 280
  • Language: English
  • Copyright: © Woodhead Publishing 2009
  • Published: March 31, 2009
  • Imprint: Woodhead Publishing
  • eBook ISBN: 9780857099358

About the Author

R F Hoskins

Affiliations and Expertise

De Montfort University, UK

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