Delta Functions

Introduction to Generalised Functions

2nd Edition - March 31, 2009

Author: R F Hoskins

eBook ISBN:

9 7 8 - 0 - 8 5 7 0 9 - 9 3 5 - 8

Delta Functions has now been updated, restructured and modernised into a second edition, to answer specific difficulties typically found by students encountering delta functions… Read more

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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.

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